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.
