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Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A measure $$\mu$$ assigns numbers $$\mu(A)$$ to certain subsets $$A$$ of a given space. More specifically, a measure is a function from a $$\sigma$$-algebra to the extended real line (i.e. it may take infinite values). A $$\sigma$$-algebra is a collection of subsets of a set $$X$$, including $$X$$ itself and closed under complements and countable unions. A measure $$\mu$$ on a $$\sigma$$-algebra $$\Sigma$$ must satisfy the following properties:

1. Nonnegativity: For every $$A\in\Sigma$$, $$\mu(A)\ge0$$.
2. Null empty set: $$\mu(\varnothing)=0$$.
3. Countable additivity: For every sequence $$(A_n)_{n=1}^\infty$$ of pairwise disjoint sets in $$\Sigma$$, $$\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n)$$.

The notion of measure is a natural generalization of the following notions:

1. Length of an interval
2. Area of a plane figure
3. Volume of a solid
4. Amount of mass contained in a region
5. Probability that an event from $$A$$ occurs, etc.

It originated in real analysis and is used now in many areas of mathematics, including geometry, probability theory, dynamical systems, functional analysis, etc.

Reference: Measure Theory