Let $(\Omega, \textit{F}_0, \mathbb{P})$ and $\textit{F} \subset \textit{F}_0$. Suppose $X \geq 0$ and $\mathbb{E}[X] = \infty$. Then there is a unique $Y \textit{F}$-measurable with $0 \leq Y \leq \infty$ so that
\begin{align*} \int_A X d\mathbb{P} = \int_A Y d\mathbb{P} \end{align*}
for all $A \in \textit{F}$
My attempt:
The book from where I took this exercise gives a hint: Let $X_M = X ∧ M$.
Then I can consider $Y_M = \mathbb{E}[X_M | \textit{F} ]$. I know that $X_M \uparrow X$, so I would like this to happen: $Y_M \uparrow \mathbb{E}[X | \textit{F}]$. I tried to use the Monotone Convergence Theorem, but I do not have that $\mathbb{E}[X] < \infty$ (I have the opposite haha).