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For increasing measurable sets, the measure of the union is the limit of the measures.
Continuity from below: Let be a measure space and let satisfy
Then
Together with continuity from above, this expresses how measures interact with countable unions and intersections of measurable sets.
A measure space is a triple where is a measurable space and is a measure on .
Measure spaces provide the basic setting for integration, convergence theorems, and statements that hold almost everywhere rather than pointwise.
Continuity from above: Let be a measure space and let satisfy
Then
This complements continuity from below and is a basic limiting property of a measure that is used repeatedly in measure-theoretic arguments.
A measure on a measurable space is a function such that and for every pairwise disjoint sequence in ,
A measure assigns “sizes” to measurable sets and turns into a measure space. Sets of measure zero are null sets, and they control notions like almost everywhere.
A measurable set in a measurable space is a subset with .
Measurable sets are precisely the subsets to which a measure assigns a value, and they determine measurable functions via preimages. The indicator function of a measurable set is a basic example of a measurable function.