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Thread: Question about notation

  1. #1
    Junior Member TheProphet's Avatar
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    Question about notation

    Let $\displaystyle X $ be a random variable on $\displaystyle (\Omega, \mathcal{A},P) $, with values in $\displaystyle (E,\mathcal{E}$, and distribution $\displaystyle P_{X}$.
    Let $\displaystyle h : (E,\mathcal{E}) \to (\mathbb{R},\mathcal{B}(\mathbb{R})) $ be measurable.

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

    Shouldn't it be $\displaystyle \mathcal{L}^{1}(E,\mathcal{E},P_{X}) $ instead? If not, why?
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  2. #2
    Super Member girdav's Avatar
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    Re: Question about notation

    Yes it's $\displaystyle \mathcal L^1(E,\mathcal E,P_X)$ (it cannot be $\displaystyle \mathcal L^1(\mathbb R,\mathcal B(\mathbb R),P_X)$ since the measure $\displaystyle P_X$ is defined on $\displaystyle \mathcal E$).
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  3. #3
    Junior Member TheProphet's Avatar
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    Re: Question about notation

    Thank you, something felt a bit off =)
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