Question about notation

**TheProphet** · August 20th 2011, 05:26 AM

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?

**girdav** · August 20th 2011, 05:49 AM

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 $\mathcal E$ ).

**TheProphet** · August 20th 2011, 06:15 AM

Thank you, something felt a bit off =)

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