Random variable divided to its conditional expectation

Question

Suppose $Y$ is positive ($Y>0$ a.s.), integrable random variable. Let $\mathcal{G}$ be sub-sigma algebra. I want to show that $$ \frac{Y}{\mathbb{E}[\,Y\mid \mathcal{G}\;]}$$ is integrable.

By the following steps I can get $$ \mathbb{E}\left[\frac{Y}{\mathbb{E}[\,Y\mid \mathcal{G}\;]}\right]=\mathbb{E}\left(\mathbb{E}\left[\frac{Y}{\mathbb{E}[\,Y\mid \mathcal{G}\;]}\mid\mathcal{G}\right]\right)=\mathbb{E}\left(\frac{\mathbb{E}[\,Y\mid \mathcal{G}\;]}{\mathbb{E}[\,Y\mid \mathcal{G}\;]}\right)=1$$ But even for he first equality above, I need integrability. Possible problems may arise when $Y$ is close to $0$, and I am unable rigorously prove the integrability.

Answer 1

Let $Y_n=\max \{Y, \frac 1 n\}$. Your argument shows that $E\left [\frac {Y_n} {E[Y_n|\mathcal G]}\right] =1$. By Fatou's Lemma we get $E \left [\frac {Y} {E[Y|\mathcal G]}\right] \leq 1$.