A set ring , also simply called a ring for short , is a special set system in measure theory and thus a set of sets. Rings and their extensions to more complex set systems such as σ-algebras play an important role in the axiomatic structure of probability theory and integration theory .
Felix Hausdorff called a set “ring” because of “an approximate analogy” to the algebraic structure of a ring in algebraic number theory . In the theory of measure today, a ring is usually understood to be a system of quantities as defined here.
The term ring used here also differs from that of a ring in the sense of algebra, but both are related.
definition
Be any set. A set system over , i.e. a set of subsets of , is called a set ring or ring over if the following properties are met:
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
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( is not empty).![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
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(Stability / isolation with regard to union ).
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(Stability / seclusion with regard to difference ).
Each set ring contains a zero element or a zero with the empty set , because it contains at least one element and therefore is .
![\ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle \ emptyset = A \ setminus A \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c75b0f1f4c6b1b11b7dc916022e507b159f6468e)
For equivalent definitions, see the relevant section below.
Examples
Power sets
Any power set is above any set
![{\ displaystyle {\ mathcal {R}} = {\ mathcal {P}} (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb562801127dca7f682668e04ec062eaad398fb6)
a set ring from a set . Because it is not empty and stable with respect to all set operations, since by definition it contains all subsets of which are also subsets of .
![{\ displaystyle T \ subseteq \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07af4d74d584df80db9bf3143e79a39e0f48b379)
![{\ displaystyle {\ mathcal {P}} (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb6ae2b499f52a9949909cccc66fdf4df64d924)
![{\ displaystyle {\ mathcal {P}} (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb6ae2b499f52a9949909cccc66fdf4df64d924)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
In particular, the power set is the largest set ring over , since it contains all subsets of .
![\ mathcal P (\ Omega)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95df45bc7d7faec1a5b88437d4b26b8a16ad108)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
The power set of the empty set is in turn the smallest set ring over , because always is.
![\ mathcal P (\ emptyset) = \ {\ emptyset \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e125ae3bb2507231e7a743d0da0298aa6d2da203)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ displaystyle \ emptyset \ subseteq \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/861401200e8d887cd4e365323978b83c46aef8bb)
System of all finite subsets
Is an arbitrary quantity and denotes the power of the quantity , then is the system
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![| A |](https://wikimedia.org/api/rest_v1/media/math/render/svg/648fce92f29d925f04d39244ccfe435320dfc6de)
![{\ displaystyle A \; (| \ emptyset | = 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ba067c7ad40937b6423e2d51668c1fec0cc749)
![{\ displaystyle {\ mathcal {R}} = \ {A \ subseteq \ Omega \ mid | A | \ in \ mathbb {N} _ {0} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d340801aa6a6b4baf5db9a09f1cc46846c552ae)
of all finite subsets of a set ring, because unions and differences of two finite sets are again finite.
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
Quantity ring of the d -dimensional figures
An important quantity ring in the application is the ring of -dimensional figures![{\ displaystyle \ mathbb {R} ^ {d}, \, d \ in \ mathbb {N},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ffc0409a02ad3239168929390a630fa3e01cbc)
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.
It consists of all sets, which can be represented as finite unions of right-open dimensional intervals , and is that of the set half-ring
![{\ displaystyle {\ mathcal {H}} = \ {[a, b) \ subset \ mathbb {R} ^ {d} \ mid a, b \ in \ mathbb {R} ^ {d} {\ text {with }} a \ leq b \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b5a84ca81b6d059855d18e6d1df1242a35381)
generated ring (see below).
properties
Stability with respect to set operations
For any two sets we always have and . Therefore, every volume ring is also stable / closed with regard to average and symmetrical difference :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![A \ cap B = A \ setminus (A \ setminus B)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6114d07903fdab2f8c8a5af611d6ffe49e1410)
![{\ displaystyle A \ bigtriangleup B = (A \ setminus B) \ cup (B \ setminus A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac7c7d817e2eb61909874cb54631da9635505aa)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
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.
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.
From the stability with respect to union, intersection and symmetrical difference it follows inductively that all finite unions as well as all non-empty, finite averages and symmetrical differences of elements of the set ring are contained in it, i.e. H. applies to all :
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
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and .![{\ displaystyle \ quad \ bigcup \ emptyset = \ emptyset \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf87e0d53da3e6ab3542ef4ffd510088c2b316a7)
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.
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.
Quantity ring with one
Since every set ring is union and average stable , it is also a set . If, as such, it also contains a single element or one , then a quantity ring with one or, for short, a ring with one .
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle \ mathrm {I}: = \ bigcup {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69894b4f8c16960e5448d685ccb250bfcdf90f3b)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
Any power set
![{\ displaystyle {\ mathcal {R}} = {\ mathcal {P}} (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb562801127dca7f682668e04ec062eaad398fb6)
a set is a set ring over with one element .
![{\ displaystyle T \ subseteq \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07af4d74d584df80db9bf3143e79a39e0f48b379)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ displaystyle \ mathrm {I} = T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90dcc43e48890e148320daf30a5be242b0e0d389)
The set system is against it
![{\ displaystyle {\ mathcal {R}} = \ {\ emptyset \} \ cup \ {\ {n_ {1}, \ ldots, n_ {m} \} \ mid n_ {1}, \ ldots, n_ {m } \ in \ mathbb {N}, \; m \ in \ mathbb {N} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10409752c2b69c2a31a640d8f8d45e48dc5685ac)
of all finite subsets of an example of a set ring without one, because .
![{\ displaystyle \ Omega = \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d00c43359504210eb7641bccb0e07be5d5303)
![{\ displaystyle \ bigcup {\ mathcal {R}} = \ mathbb {N} \ notin {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20159edd4565170afd81a63cff9f2af83f20349b)
Relationship to the ring in the sense of algebra
The triple with the set ring is a ring in the sense of algebra and the empty set is its zero element . If a lot of ring with unity, is also the identity of .
![{\ displaystyle ({\ mathcal {R}}, \ bigtriangleup, \ cap)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084a2866b9b37eca316b0fec3b17b1b21d0c063b)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle \ mathrm {I} = \ bigcup {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6db227cd29fcbaecae230f4b73046155cb071f4a)
![{\ displaystyle ({\ mathcal {R}}, \ bigtriangleup, \ cap)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084a2866b9b37eca316b0fec3b17b1b21d0c063b)
Conversely, if a system of sets is such that a ring is in the sense of algebra, then there is always a set ring because of and for all .
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle ({\ mathcal {R}}, \ bigtriangleup, \ cap)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084a2866b9b37eca316b0fec3b17b1b21d0c063b)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle A \ cup B = (A \ bigtriangleup B) \ bigtriangleup (A \ cap B) \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a97cb7ec6439f37a1c18ce70d410790058b77ec1)
![{\ displaystyle A \ setminus B = A \ bigtriangleup (A \ cap B) \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c2505974b0e928c7717d84e318e0cd573cb92f)
![{\ displaystyle A, B \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96774c52aaf8069910c7eb3f59040fed9c50c882)
So that each set ring can be represented as a ring in the sense of algebra, it must not be empty, because the empty set cannot contain a zero element and therefore cannot be a carrier set of a ring in the sense of algebra.
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7)
Equivalent Definitions
If is a system of subsets of and if are sets, then because of and the following two statements are equivalent :
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![A \ cap B = A \ setminus (A \ setminus B)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6114d07903fdab2f8c8a5af611d6ffe49e1410)
![A \ setminus B = A \ setminus (A \ cap B)](https://wikimedia.org/api/rest_v1/media/math/render/svg/24ee7ab5da0232f500480c470cc42e11339b038a)
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.
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and if also .![B \ subseteq A](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8124cb68686ede7083aa2a5a821f262eb62954)
![{\ displaystyle A \ setminus B \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/047c9dfbb3f004425f7ce9ff8366c86b7757e5a4)
If besides , then because of and as well as for every set with are also equivalent:
![{\ mathcal R} \ neq \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c92c3467a1a7bbdda4ebe261fedf64a2525435b)
![{\ displaystyle A \ setminus B = (A \ cup B) \ bigtriangleup B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b498cab221d77099b9782e9a5f75ca733deee275)
![A \ cup B = (A \ setminus B) \ cup B](https://wikimedia.org/api/rest_v1/media/math/render/svg/30f6d3f12c63387db3dbe273f73abbde74c104c9)
![A \ cup B = C \ setminus ((C \ setminus A) \ cap (C \ setminus B))](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11fb9e264f704cb676bf829b50f0f2872c9fedc)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![A \ cup B \ subseteq C](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdb1727a9bbdd71e93d93c8ac77260b8ac163c8)
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is a quantity ring.
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is a differential stable set .
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is a union- stable set half-ring .
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is stable with respect to symmetrical difference and average .![\ bigtriangleup](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0cf1edbbf9956e17cc233dba05842b5ea1291)
![\ cap](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3)
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is an Abelian group and is a semigroup .![({\ mathcal R}, \ cap)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aac7b5d6b4da8d2c22c562e360f7ba508894f30)
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is a ring in the sense of algebra with addition and multiplication .![\ bigtriangleup](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0cf1edbbf9956e17cc233dba05842b5ea1291)
![\ cap](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4e886e6f5a28a33e073fb108440c152ecfe2d3)
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is an idempotent ( commutative ) ring in the sense of algebra.
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is stable with respect to symmetric difference and union .![\ bigtriangleup](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0cf1edbbf9956e17cc233dba05842b5ea1291)
![\ cup](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ff7d0293ad19b43524a133ae5129f3d71f2040)
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and if there is a with .![A \ cap B = \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0814f1814eb415cc84354bf261d7f9ed62367d1)
![C \ in {\ mathcal R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf04a2ed20ca3a7f191f563e4a0c22e021395f9)
![A \ cup B \ subseteq C](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdb1727a9bbdd71e93d93c8ac77260b8ac163c8)
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and there is a with .![C \ in {\ mathcal R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf04a2ed20ca3a7f191f563e4a0c22e021395f9)
![A \ cup B \ subseteq C](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdb1727a9bbdd71e93d93c8ac77260b8ac163c8)
Operations with rings
Cut of rings
The intersection of two quantity rings and is always a ring again. Because are , so are and , so as well . Thus, also in , the cut is consequently stable with respect to union. The stability with regard to the difference follows analogously.
![{\ displaystyle {\ mathcal {R}} _ {1} \ cap {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f91de42e0d27c412e55f98f9686dfe55b2e81d2a)
![{\ displaystyle {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120b472139db67f914386100f3076394525d60f9)
![{\ displaystyle {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce80eda8a70b1fc882429d301f7d0c5d13af30d)
![{\ displaystyle A, B \ in {\ mathcal {R}} _ {1} \ cap {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31080b1c2e18271845492aac06e72e0f52886f4f)
![{\ displaystyle A, B \ in {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3896c0470af08751f60e1946a3010051fa347fde)
![{\ displaystyle A, B \ in {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2cdd470e775220dfa0b65f5519054038bedbe4e)
![{\ displaystyle A \ cup B \ in {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d7a5530e1953f88f5d3938c4eb41cd75a468b5)
![{\ displaystyle A \ cup B \ in {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9beb680adced3f30a240dee246b20c4e89e2f530)
![A \ cup B](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169)
![{\ displaystyle {\ mathcal {R}} _ {1} \ cap {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f91de42e0d27c412e55f98f9686dfe55b2e81d2a)
The statement also applies to the intersection of any number of set rings, since the above argument can then be extended to all of these rings. Thus:
If there is an arbitrary index set and if they are all over the same basic set for quantity rings , then the intersection of all these rings is again a quantity ring over :
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![{\ displaystyle {\ mathcal {R}} _ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4e99f70ce55a732f202d636b5fdc6a28e13440)
![i \ in I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
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.
Association of Rings
The union of two set rings and is generally no longer a set ring. For example, consider the two rings
![{\ displaystyle {\ mathcal {R}} _ {1} \ cup {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76dab2f4c80ca563aad30393c52d7f471b2e962c)
![{\ displaystyle {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120b472139db67f914386100f3076394525d60f9)
![{\ displaystyle {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce80eda8a70b1fc882429d301f7d0c5d13af30d)
![{\ displaystyle {\ mathcal {R}} _ {1} = \ {\ emptyset, \ {1 \}, \ {2,3 \}, \ {1,2,3 \} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d448d3d517b9fb551b489f273973d9a09c4339ab)
such as
-
,
so is
-
.
However, this system of sets is neither union-stable, since it does not contain, nor is it differential- stable because it does not contain, and thus also no set ring.
![{\ displaystyle \ {1 \} \ cup \ {2 \} = \ {1,2 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff706d9cd2b535c8dfb6dd1e188db0fcc78dc08c)
![{\ displaystyle \ {1,3 \} \ setminus \ {1 \} = \ {3 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4666f3922d1eb0372a59140991abff0beea3b08e)
Product of rings
Let there be a set system over and a set system over . The direct product of and is defined as the system of sets
![{\ mathcal {S}} _ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dbf01a888be8dd4184319517189418a78a1a59)
![{\ displaystyle \ Omega _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95a2e6dbc4b6f2d03b3397691609aa9544230f7)
![{\ mathcal {S}} _ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0470c26679126290f674631a2022e5b71e3df7)
![{\ displaystyle \ Omega _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789466adc27ae2abb1feb4e8a00d374d8c6521da)
![{\ mathcal {S}} _ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dbf01a888be8dd4184319517189418a78a1a59)
![{\ mathcal {S}} _ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0470c26679126290f674631a2022e5b71e3df7)
![{\ displaystyle {\ mathcal {S}} _ {1} \ boxdot {\ mathcal {S}} _ {2}: = \ {A \! \ times \! B \ mid A \ in {\ mathcal {S} } _ {1}, \; B \ in {\ mathcal {S}} _ {2} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e8b855e065914e4fd771e576ca7385ef10a9ea)
over .
![{\ displaystyle \ Omega _ {1} \ times \ Omega _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/540f31fd90794ea8aa0d9f3946afa650142176eb)
However, the direct product of two set rings is generally no longer a set ring, but only a set half- ring .
Consider the power set ring as a counterexample
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,
so the system of sets contains the sets
![{\ displaystyle {\ mathcal {R}} \ boxdot {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8dd9dccfb595a38c29de1b0c8220efee2527d2)
![{\ displaystyle A = \ {1 \} \ times \ {1 \} = \ {(1,1) \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99fe50798b6c7f42660c212b44f0201630441e60)
and
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.
The amount
![{\ displaystyle B \ setminus A = \ {(1,2), (2,1), (2,2) \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6de0a8f25ed6538507622ea4c1e34c68d0c14d3)
is not included in, however , because it cannot be represented as the Cartesian product of two sets of . Thus, the direct product is not differential stable and therefore also not a quantity ring.
![{\ displaystyle {\ mathcal {R}} \ boxdot {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8dd9dccfb595a38c29de1b0c8220efee2527d2)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle {\ mathcal {R}} \ boxdot {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8dd9dccfb595a38c29de1b0c8220efee2527d2)
The ring product of two sets of rings over and over is therefore defined as their tensor product![{\ displaystyle {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120b472139db67f914386100f3076394525d60f9)
![{\ displaystyle \ Omega _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95a2e6dbc4b6f2d03b3397691609aa9544230f7)
![{\ displaystyle {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce80eda8a70b1fc882429d301f7d0c5d13af30d)
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,
so that this is again a quantity ring above , namely the ring generated by (see below).
![{\ displaystyle \ Omega _ {1} \ times \ Omega _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/540f31fd90794ea8aa0d9f3946afa650142176eb)
![{\ displaystyle {\ mathcal {R}} _ {1} \ boxdot {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b34626bd2a3ba1a87364d150ab7b4a3f95404abe)
Trace of a ring
The trace of a ring over in a set , so the set system
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ displaystyle T \ subseteq \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07af4d74d584df80db9bf3143e79a39e0f48b379)
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,
is always a quantity ring over and over .
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
Creation of rings
Since any sections of quantity rings are back rings (s. O.), Can be for any amount system over by
![{\ mathcal {S}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ displaystyle \ varrho ({\ mathcal {S}}): = \ bigcap \; \ {{\ mathcal {R}} \ mid {\ mathcal {R}} {\ text {is a ring over}} \ Omega {\ text {with}} {\ mathcal {S}} \ subseteq {\ mathcal {R}} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa10ebd8e1d87c93b3e6b1b3e2533ad615fb4a10)
define an envelope . By definition, this is the smallest volume ring above that contains the system of quantities and is called the ring generated by .
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ mathcal {S}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2302a18e269dbecc43c57c0c2aced3bfae15278d)
Sometimes the generated ring can be specified directly. Thus the ring created by a set half -ring is of the shape
![{\ mathcal {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f)
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.
An explicit example of this form is the above example of the set ring of -dimensional figures
.
The same applies to the product of two quantity rings discussed above and :
![{\ displaystyle {\ mathcal {R}} _ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120b472139db67f914386100f3076394525d60f9)
![{\ displaystyle {\ mathcal {R}} _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce80eda8a70b1fc882429d301f7d0c5d13af30d)
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.
Related set systems
Hierarchy of the quantity systems used in measure theory
Generalizations
- Quantity half-ring
Every set ring is a (union- stable ) set half- ring , but not every set half-ring is also a set ring:
Because the quantity half-ring
![{\ displaystyle {\ mathcal {H}} = \ {\ emptyset, \ {1 \}, \ {2 \}, \ {3 \}, \ {1,2,3 \} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4916f1cd60ee49dea2570acba33700b8254166f6)
is not a set ring because it is neither union nor difference stable.
![{\ mathcal {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f)
- Mass association
A set ring is always a (differential-stable) set , but not every set is a set ring:
The mass association
![{\ displaystyle {\ mathcal {V}} = \ {\ {1 \} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1348f8460f07c5269e2f49e81775a98626df988e)
is not a quantity ring because it is not differential stable.
![\ mathcal V](https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6)
Special quantity rings
- Set algebra
A set ring over a set with is called a set algebra over . So every set algebra is a set ring with the one , but not every set ring is a set algebra.
![\ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\ Omega \ in \ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcb18d1cac31b6b1ab192a64a4c11f3cc4359a9)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![\ mathcal A](https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8)
![{\ displaystyle \ Omega = \ bigcup {\ mathcal {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f95fe17b3fd7a49600fe9bbd2e99974bd94df7fe)
So is the quantity ring
![{\ displaystyle {\ mathcal {R}} = {\ mathcal {P}} (\ {1 \}) = \ {\ emptyset, \ {1 \} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3615ca3c93d8e1668337c4921c4e071e6b5dee72)
no set algebra over the basic set , there . If, on the other hand, one takes its one as the basic set, then a set algebra is and thus is over .
![{\ displaystyle \ Omega = \ {1,2 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b095a095061e3d68081b187f04febff99ca40c45)
![{\ displaystyle \ Omega \ notin {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/355a7f6d7eeb9bd934756aea3f04769ab96cc5c1)
![\ mathrm {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7a69180f25bbb4c73e091f97c7c5f9941ed17b)
![{\ displaystyle \ mathrm {I} = \ {1 \} \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80131180ffac5f9541030e021dc86b39f0c44547)
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![\ mathrm {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7a69180f25bbb4c73e091f97c7c5f9941ed17b)
The assumed basic set is therefore essential for the concept of set algebra.
- δ ring
A set ring that is closed with respect to a countable number of sections is called a δ-ring .
- σ ring
A ring of sets that is closed with respect to a countable number of unions is called a σ-ring .
- Monotonous classes
Every ring that is a monotonic class is a σ-ring (and therefore also a δ-ring). Because if everyone is for , so is the properties of the ring
![\ mathcal R](https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f)
![{\ displaystyle A_ {1}, \ ldots, A_ {n} \ in {\ mathcal {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71308558a62d18a37379f21b6eff5932ff738434)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
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However, the sets form a monotonically increasing sequence of sets , which is why its limit value is due to the properties of the monotonous class
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is thus closed with respect to countable unions. Thus the monotonic class generated by a ring is always a σ-ring.
See also
literature
Individual evidence
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↑ Felix Hausdorff: Fundamentals of set theory . Veit & Comp., Leipzig 1914, p. 14 . Hausdorff referred to the association as "sum".
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↑ Hausdorff called such a "body" ( Fundamentals of set theory . P. 15) .
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↑
Peter Eichelsbacher: Probability Theory . Ruhr University Bochum, S. 5 f . ( [1] [PDF; accessed October 30, 2018] Lecture notes for the winter semester 2016/17).