Questions tagged [measure-theory]

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