Random Variable

Measurable function

Let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces where $E,F,$ are sets and $\mathcal{E}$ and $\mathcal{F}$ are $\sigma$-field on $E$ and $F$, respectively. Then a function $f$ is measurable function if every pre-image of $B \in \mathcal{F}$ is measurable.

\[f^{-1}(B) = \{ x \in E: f(x) \in B \} \in \mathcal{E}\]

Random variable

Let $(\Omega, \mathcal{A}, \mathbb{P})$ and $(E, \mathcal{E})$ be a probability space and measurable space, respectively. A $\mathcal{A}/\mathcal{E}$ measurable function $X$ is a random variable, satisfying the following:

\[X^{-1}(B) = \{ \omega \in \Omega: X(\omega) \in B \} \in \mathcal{A} \text{, for evey } B \in \mathcal{E}.\]

If we set $E = \mathbb{R}$, we take $\mathcal{E}$ to be the smallest $\sigma$-field, that contains all the open subsets, which is called the Borel $\sigma$-field.

Distribution

Let $X$ be a random variable on measurable space $(E, \mathcal{E})$. The distribution of $X$ is defined as

\[\mu(A) = \mathbb{P}(X \in A) = \mathbb{P}(X^{-1}(A)) = \mathbb{P}(\{ \omega \in \Omega: X(\omega)\in A\}).\]

The probability space $(\Omega, \mathcal{A}, \mathbb{P})$ is often called background probability space and the measure space $(E, \mathcal{E}, \mu)$ is called the induced probability space.