Monotonous sequence of quantities
A monotonous sequence of sets is a special sequence of sets in which special inclusion relationships apply. If a set with a smaller index is always contained in a set with a larger index, then the sequence is called a monotonically increasing set sequence . If a set with a smaller index always contains a set with a larger index, the sequence is called a monotonically decreasing sequence of sets . Monotonous set sequences can be understood as a special case of a monotonous mapping .
definition
A sequence of sets is called
- Monotonically increasing or monotonically increasing , if applies.
- Monotonously decreasing if applies.
- Monotone if it is either monotonically increasing or monotonically decreasing.
Sometimes there is also the designation of a monotonically increasing set sequence or a monotonically decreasing set sequence .
Examples
- The sequence of quantities defined by
- is a monotonically increasing set sequence, since every set contains all elements of the set .
- The sequence is monotonically increasing. This follows directly from the monotony of the real sequence .
- Likewise, the sequence is monotonically decreasing.
properties
- Every monotonically increasing sequence of sets converges , then it is
- .
- You then also write .
- Every monotonically decreasing sequence of sets converges, then it is
- .
- You then also write .
use
Monotonic set sequences are used in measure theory , for example , to define set systems like monotonic classes .
literature
- Jürgen Elstrodt: Measure and integration theory . 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .