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Math Help - Question about notation

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

    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?
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  2. #2
    Super Member girdav's Avatar
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    Re: Question about notation

    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).
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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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