How can we define conditional expectation of a positive but non integrable random variable $Y$?

Question

How can we define conditional expectation of a positive but non integrable random variable $Y$ ?

I have no idea on how to proceed since the Hilbert space construction and the cutting off procedure requires atleast that $Y \in L^1$ and we need integrability even in the proof of existence of conditional expectation via the Radon Nikodym theorem. I was trying to think about the following problem in Probability and Measure by Billigsley

Even a reference would be highly appreciated.

Answer 1

For $X$ a nonnegative random variable, you can define $\mathbb E[X\vert\mathcal G]=\lim_{n\to+\infty}\mathbb E[X\wedge n\vert G]$.