Measurable

Let $({\Omega }_1,{\mathcal{F}}_1)$ and $({\Omega }_2,{\mathcal{F}}_2)$ be two measurable spaces. A function $f:{\Omega }_1\to {\Omega }_2$ is $({\mathcal{F}}_1,{\mathcal{F}}_2)$-measurable if the preimage of any measurable set ($B\in {\mathcal{F}}_2$) is measurable $f^{-1}(B)\in {\mathcal{F}}_1$.

• A Random Variable is a $(\mathcal{F},\mathcal{B})$-measurable function where $\mathcal{F}$ is the Sigma Field on the Probability Space and $\mathcal{B}$ is the Borel Sigma Field on the real line.
• A continuous function $f:\mathbb{R}\to \mathbb{R}$ is Borel-measurable.
• The composition of measurable functions is measurable.
  • Specifically, given a continuous function $f$ and Random Variable $X$, $f(X)$ is a Random Variable.
  • Further, given a continuous multivariate function $f:{\mathbb{R}}^n\to \mathbb{R}$ and Random Variables $X_1,\dots ,X_n$, $f(X_1,\dots ,X_n)$ is a Random Variable.
    • In particular, the sum and product of Random Variables are Random Variables.