• Aug 20th 2011, 06:26 AM
TheProphet
Let $X$ be a random variable on $(\Omega, \mathcal{A},P)$, with values in $(E,\mathcal{E}$, and distribution $P_{X}$.
Let $h : (E,\mathcal{E}) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ be measurable.

We have that $h(X) \in \mathcal{L}^{1}(\Omega,\mathcal{A},P)$ if and only if $h \in \mathcal{L}^{1}(\mathbb{R},\mathcal{B}(\mathbb{R}) ,P_{X})$.

Shouldn't it be $\mathcal{L}^{1}(E,\mathcal{E},P_{X})$ instead? If not, why?
• Aug 20th 2011, 06:49 AM
girdav
Yes it's $\mathcal L^1(E,\mathcal E,P_X)$ (it cannot be $\mathcal L^1(\mathbb R,\mathcal B(\mathbb R),P_X)$ since the measure $P_X$ is defined on $\mathcal E$).