definition of measure over \(\sigma\)-algebra of set
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of measurable space.
Target Context
- The reader will have a definition of measure over \(\sigma\)-algebra of set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( (S, A)\): \(\in \{ \text{ the measurable spaces }\}\)
\(*\mu\): \(: A \to [0, + \infty]\)
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Conditions:
\(\mu (\emptyset) = 0\)
\(\land\)
\(\forall s: \mathbb{N} \to A \text{ such that } \forall n_1, n_2 \in \mathbb{N} \text{ such that } n_1 \neq n_2 (s (n_1) \cap s (n_2) = \emptyset) (\mu (\cup_{j \in \mathbb{N}} s (j)) = \sum_{j \in \mathbb{N}} \mu (s (j)))\)
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2: Note
There are also 'signed measure over \(\sigma\)-algebra of set' and 'complex measure over \(\sigma\)-algebra of set'.
