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.
  • In addition, leave
    $$ f: mathbb {R} ^ 2 rightarrow mathbb {R} $$
    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?

Thank you in advance for any advice!