More precisely let $(X,Y)$ is a pair of continuous random variables with joint density function $f(x,y)$ and we assume $\mathbb{E}(|Y|) < +\infty$. Define $$H(X) = \int\limits_{-\infty}^\infty \frac{f(x,y)}{f_X(x)}y dy,$$ where $f_X(x) = \int\limits_{-\infty}^\infty f(x,y) dy$.
For $H(X)$ to satisfy the definition we have to show that
- $\mathbb{E}(|H(X)|) < +\infty$,
- $H(X)$ is $\sigma(X)$-measurable and
- $\mathbb{E}(Y \cdot I_A) = \mathbb{E}(H(X) \cdot I_A)$, for every $A \in \sigma(X)$, where $I_A$ is the indicator of $A$.
This is a homework assignment for me, so I only need some hints, not a full solution.
Hint on third bullet:
On base of $H\left(x\right)f_{X}\left(x\right)=\int f\left(x,y\right)ydy$ we find for suitable functions $g$:
$$\mathbb{E}\left[H\left(X\right)g\left(X\right)\right]=\int H\left(x\right)g\left(x\right)f_{X}\left(x\right)dx=\int\int f\left(x,y\right)yg\left(x\right)dydx=\mathbb{E}\left[Yg\left(X\right)\right]$$