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?