# Is the conditional expectation of the conditional expectation equal to the conditional expectation [closed]

Does $$\int E_{Y\lvert X}[Y\lvert X=x] p(y\lvert x)dy = E_{Y\lvert X}[Y\lvert X=x]$$

In other words is the conditional expectation of the conditional expectation equal to the conditional expectation?

We know that there exists a measurable map $$\varphi$$ such that almost surely, $$E[Y\vert X]=\varphi(X)$$. So $$E[E[Y\vert X]\vert X]=E[\varphi(X)\vert X]=\varphi(X)=E[Y\vert X].$$
For random variables $$X,Y$$ (defined on same probability space) we have under suitable conditions the random variable $$\mathbb E[Y\mid X]$$. It is measurable wrt $$\sigma(X)$$ and can be written as $$f(X)$$ where $$f:\mathbb R\to\mathbb R$$ is a Borel-measurable function. Then $$f(x)=\mathbb E[Y\mid X=x]$$ and consequently:$$\int\mathbb E[Y\mid X=x]p(y|x)dy=\int f(x)p(y|x)dy=f(x)\int p(y|x)dy=f(x)=\mathbb E[Y\mid X=x]$$