s-finite measure

A certain class of measures in measure theory , a branch of mathematics, is called s-finite measures or s-finite measures . They can be represented as a countable sum of finite measures and thus allow the generalization of certain proofs. The s-finite measures are similar to the σ-finite measures , but should not be confused with them.

definition

A measuring room is given . Then a measure in this measurement space is called an s-finite measure if there is a countable sequence of finite measures such that ${\ displaystyle (X, {\ mathcal {A}})}$ ${\ displaystyle \ mu}$ ${\ displaystyle (\ nu _ {n}) _ {n \ in \ mathbb {N}}}$

{\ displaystyle \ mu = \ sum _ {i = 1} ^ {\ infty} \ nu _ {i}}

applies.

example

The Lebesgue measure is an s-finite measure. To do this, define ${\ displaystyle \ lambda}$

{\ displaystyle I_ {n}: = [- n, n]}

and

{\ displaystyle B_ {n}: = I_ {n} \ setminus I_ {n-1}}

.

If the Lebesgue measure is restricted to the quantity , then the measures are ${\ displaystyle \ lambda | _ {B}}$ ${\ displaystyle B}$

{\ displaystyle \ nu _ {n} = \ lambda | _ {B_ {n}}}

all finite and add up due to their construction . ${\ displaystyle \ lambda}$

properties

Relation to σ-finiteness

Every σ-finite measure is always s-finite. Because if σ-finite and measurable disjoint sets are required for all as in the definition of σ-finiteness, then finite measures are to be added up again as in the example above . Conversely, not every s-finite measure is also σ-finite. Considering as measurement space, the amount , provided with the power set as σ-algebra and the dimensions defined all as the counting measure on , so ${\ displaystyle \ nu}$ ${\ displaystyle B_ {1}, B_ {2}, B_ {3}, \ dots}$ ${\ displaystyle \ mu (B_ {i}) <\ infty}$ ${\ displaystyle i \ in \ mathbb {N}}$ ${\ displaystyle \ nu _ {n}: = \ mu | _ {B_ {n}}}$ ${\ displaystyle \ mu}$ ${\ displaystyle X = \ {a, b \}}$ ${\ displaystyle \ nu _ {n}}$ ${\ displaystyle X}$

{\ displaystyle \ mu: = \ sum _ {i = 1} ^ {\ infty} \ nu _ {i}}

by construction s-finite. But is not σ-finite, because it is ${\ displaystyle \ mu}$

{\ displaystyle \ mu (\ {a \}) = \ sum _ {i = 1} ^ {\ infty} \ nu _ {i} (\ {a \}) = \ sum _ {i = 1} ^ { \ infty} 1 = \ infty}

,

the case for follows analogously. ${\ displaystyle \ {b \}}$

equivalence

Every s-finite measure is equivalent to a probability measure . That means that there is a measure with such that . Here means that and , that is, it is absolutely continuous with respect to and absolutely continuous with respect to . Because if finite measures are required as in the definition of s-finiteness, then a possible one is given by ${\ displaystyle \ mu}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ nu (X) \ leq 1}$ ${\ displaystyle \ nu \ sim \ mu}$ ${\ displaystyle \ nu \ sim \ mu}$ ${\ displaystyle \ nu \ ll \ mu}$ ${\ displaystyle \ nu \ gg \ mu}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ nu}$ ${\ displaystyle (\ nu _ {i}) _ {i \ in \ mathbb {N}}}$ ${\ displaystyle \ nu}$