Specific question about the proof Law of total expectation

Question

Can someone please explain me just the yellow part? Why is the first step is equal to the second?

Proof:

Answer 1

I'm not sure if this will help with what is confusing, Amit, but I think it might.

To get from $E[E[X|Y]]$ to $E\left[\sum_x x \, P(X=x|Y)\right]$ we just replace the inner expectation with the definition of conditional expectation applied to $E[X|Y]$.

A conditional expectation is actually a random variable, though. In this case, it is a function from possible values of $Y$ to simple (non-conditional) expectations. For each value $Y=y$, we have a different simple expectation. Think of this as the expectation of $X$, restricted to the part of the probability space where $Y=y$. What is the expected value of $X$ within this subset of outcomes, i.e. when we fix the value of $y$? For each such $y$, we have a different (simple) expectation. That's why a conditional expectation is a function, a random variable.

To get to the next line,

$$\sum_y\left[\sum_x x P(X=x|Y=y)\right]P(Y=y)$$

we are simply taking the (non-conditional) expectation of what's inside the outer expectation operator's scope. That is, we are taking the expectation of a random variable that is a function of $y$, $E[X|Y=y]=\sum_x x\,P(X=x|Y=y)$. Notice that while $x$ is bound by the summation operator, $y$ is not. This notation highlights the role of $y$ in the conditional expectation, treated as a function of $y$.

The second line of the proof, shown indented above, is precisely the application of the expectation operator to the random variable $E[X|Y=y]$.