What is the exact meaning of the subscript notation $\mathbb{E}_X[f(X)]$ in conditional expectations in the framework of measure theory ? These subscripts do not appear in the definition of conditional expectation, but we may see for example in this page of wikipedia. (Note that it wasn't always the case, the same page few months ago).
What should be for example the meaning of $\mathbb{E}_X[X+Y]$ with $X\sim\mathcal{N}(0,1)$ and $Y=X+1$ ?
In an expression where more than one random variables are involved, the symbol $E$ alone does not clarify with respect to which random variable is the expected value "taken". For example
$$E[h(X,Y)] =\text{?} \int_{-\infty}^{\infty} h(x,y) f_X(x)\,dx$$ or $$E[h(X,Y)] = \text{?} \int_{-\infty}^\infty h(x,y) f_Y(y)\,dy$$
Neither. When many random variables are involved, and there is no subscript in the $E$ symbol, the expected value is taken with respect to their joint distribution:
$$E[h(X,Y)] = \int_{-\infty}^\infty \int_{-\infty}^\infty h(x,y) f_{XY}(x,y) \, dx \, dy$$
When a subscript is present... in some cases it tells us on which variable we should condition. So
$$E_X[h(X,Y)] = E[h(X,Y)\mid X] = \int_{-\infty}^\infty h(x,y) f_{h(X,Y)\mid X}(h(x,y)\mid x)\,dy $$
Here, we "integrate out" the $Y$ variable, and we are left with a function of $X$.
...But in other cases, it tells us which marginal density to use for the "averaging"
$$E_X[h(X,Y)] = \int_{-\infty}^\infty h(x,y) f_{X}(x) \, dx $$
Here, we "average over" the $X$ variable, and we are left with a function of $Y$.
Rather confusing I would say, but who said that scientific notation is totally free of ambiguity or multiple use? You should look how each author defines the use of such symbols.
I just want to add a follow-up to Alecos' great answer. Sometimes it doesn't matter the exact R.V. (or set of RV) the expectation is over. For instance,
$$ E_{X\sim P(X)} [X] = E_{X\sim P(X,Y)}[X] $$
In your particular question, I suspect that because you are given $h(X,Y)$ is linear in X and Y, then you will break it up into the "marginal" expectations $E_X[X]$ and $E_X[Y]$ (and then swap in $Y = X + 1$)
I will add to Aleco's answer by mentioning a few different notations which are common in the literature of statistics and machine learning. The most common interpretation seems to be the second one in Aleco's answer, i.e. we use the marginal pmf/pdf of the mentioned variable, and sum/integrate over all possible values of the mentioned variable.
The only text I found which explicitly mentions this notation is the PRML book [1], which mentions:
Sometimes we will be considering expectations of functions of several variables, in which case we can use a subscript to indicate which variable is being averaged over, so that for instance
$$ E_{X} [h(X,Y)] $$ denotes the average of the function $H(X, Y)$ with respect to the distribution of X. Note that $E_{X} [h(X,Y)] $ will be a function of Y.
As I understand this:
If X & Y are discrete: $$ g(Y) = E_{X} [h(X,Y)] = \sum_{x \in Range(X)} h(x, Y) p_{X}(x) $$
If X & Y are continuous $$ g(Y) = E_{X} [h(X,Y)] = \int_{-\infty}^\infty h(x, Y) f_{X}(x) \, dx $$
Notice that both of these are functions of $Y$, since we are "averaging" over all possible values of $X$ and multiplying by their respective probabilities.
Another common subscript is $X \sim D$, which means the same thing. By using this subscript, the author is trying to tell us that the r.v. X is distributed according to some probability distribution $D$ (pmf or pdf), and that the distribution is an important part of the equation:
If X & Y are discrete: $$ g(Y) = E_{X\sim D} [h(X,Y)] = \sum_{x \in Range(X)} h(x, Y) D(x) $$ where $D(x) = p_{X}(x)$ is the marginal pmf of X.
If X & Y are continuous: $$ g(Y) = E_{X\sim D} [h(X,Y)] = \int_{-\infty}^\infty h(x, Y) D(x) \, dx $$ where $D(x) = f_{X}(x)$ is the marginal pdf of X.
[1] Pattern Recognition and Machine Learning, Bishop, pg 20 (equation 1.36)
