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world:sigmafield


2017/10/07 23:39 ·

# A useful result

Exercise Let $(\Omega, \mc{F}, \PP)$ be a probability space and let $X,Y$ be two random variables such that $Y$ is $\sigma(X)$-measurable. Then there exists a $\mc{B}(\rset)/\mc{B}(\rset)$-measurable function $h: \rset \to \rset$ such that $Y=h(X)$.

$\bproof$ Using the decomposition $Y=Y^+-Y^-$ if necessary, we now assume that $Y \in \rset^+\cup \{\infty\}$.

By assumption, for every $(k,n) \in \nset \times \nset^*$, there exist $B_{k,n}, B_n \in \mc{B}(\rset)$ such that $Y^{-1}\left[\frac{k}{n},\frac{k+1}{n}\right)=X^{-1}(B_{k,n})$ and $Y^{-1}\left[n,\infty\right)=X^{-1}(B_n)$. Then, \begin{align*} Y_n(\omega)&= \sum_{k=0}^{n^2-1} \frac{k}{n}\ \indi{Y^{-1}\left[\frac{k}{n},\frac{k+1}{n}\right)}(\omega)+n \indi{Y^{-1}\left[n,\infty\right)}(\omega)=\sum_{k=0}^{n^2-1} \frac{k}{n}\ \indi{X^{-1}(B_{k,n})}(\omega)+n\indi{X^{-1}(B_{n})}(\omega)=\underbrace{\lr{\sum_{k=0}^{n^2-1} \frac{k}{n}\ \indi{B_{k,n}}+n \indi{B_n}}}_{h_n} \circ X(\omega). \end{align*} Setting $h=\limsup_n h_n$, we have $h$ is $\mc{B}(\rset)/\mc{B}(\rset)$-measurable as a limit of $\mc{B}(\rset)/\mc{B}(\rset)$-measurable functions. Moreover, $$Y=\limsup_n Y_n=\limsup_n (h_n \circ X)=(\limsup_n h_n) \circ X= h \circ X$$ $\eproof$

world/sigmafield.txt · Last modified: 2022/03/16 07:40 (external edit)