Measures

This page is under construction.

Definition

Let $X$ be a set equipped with a $\sigma$-algebra ${\mathcal {M}}$. A measure on $(X,{\mathcal {M}})$ (also referred to simply as measure on $X$ if $M$ is understood) is a function $\mu :{\mathcal {M}}\rightarrow [0,\infty ]$ that satisfies the following criteria:

1. $\mu \left(\emptyset \right)=0$,
2. Let $\left\{E_{k}\right\}_{k=1}^{\infty }$ be a disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu \left(E_{k}\right)$.

Properties

1. Countable Additivity: Let $\left\{E_{k}\right\}_{k=1}^{n}$ be a finite disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{n}E_{k}\right)=\sum _{k=1}^{n}\mu \left(E_{k}\right)$. This follows directly from the defintion of measures by taking $E_{n+1}=E_{n+2}=...=\emptyset$.

Examples

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References

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