# pr.probability – Concrete substitution value in conditional expectation

• Let $$( Omega, mathcal {G}, mathbb {P})$$ to be a probability space.
• Let $$X, Y: Omega rightarrow mathbb {R}$$ to be random variables.
$$f: mathbb {R} ^ 2 rightarrow mathbb {R}$$
be a $$mathcal {B} ( mathbb {R} ^ 2) / mathcal {B} ( mathbb {R})$$measurable function
such as, for all $$y in mathbb {R}$$, random variables $$f (X, y)$$ and
$$f (X, Y)$$ have finite expectations.

Now let $$y in mathbb {R}$$ to be arbitrary. According to the above assumptions, the expected value $$mathbb {E}[f(X,y)]$$ and one $$mathbb {P}$$– single conditional expectation $$mathbb {E}[f(X,Y) mid Y]$$ exist.

Moreover, since $$mathbb {E}[f(X,Y) mid Y]$$ is $$sigma (Y) / mathcal {B} ( mathbb {R})$$-measurable, there is a $$mathbb {P} _Y$$-unique $$mathcal {B} ( mathbb {R}) / mathcal {B} ( mathbb {R})$$measurable function
$$varphi: mathbb {R} rightarrow mathbb {R}$$
such as $$varphi (Y) = mathbb {E}[f(X,Y) mid Y]$$.

Under what circumstances $$varphi$$ can be chosen as
$$varphi (y) = mathbb {E}[f(X,y)]$$
and why?