Vtyjkyuk

Let's say I have a set $Omega$ with a probability $P$ on its subsets (the fact that it could be undefined on some subset is not important here, let's not think about that). I have a random variable $(X,Y): Omega to mathbbR^2$, and a function $phi: mathbbR^2 to mathbbR^2$. Then, I define the probability induced by the random variable as $P_X,Y(A)=P((X,Y)^-1(A))$, with $A subseteq mathbbR^2$. A well known result tells me that (given $omega=(x,y) in Omega$): $$int_Omega phi((X,Y)(omega))dP=int_mathbbR^2 phi(x) dP_X,Y$$

Consequently, taking $phi =x+y$, I have that $E[(X,Y)]= int_Omega (X,Y)(omega) dP=sum_x_j,y_i in Omega (X,Y)(x_i,y_j) p_x,y(x_i,y_j)$ if $Omega$ is countable, where $p_x,y$ is the probability that $(X,Y)(omega)=(x_i,y_j)$.

What I don't get is why it is that $E[X+Y]=E[X]+E[Y]$ when $X$ and $Y$ are integrable. For instance, if I had a random variable $X+Y$ with $X$ and $Y$ that assume all positive integer values, I don't get why $$E[X+Y]=sum_n=1^infty n p_x,y(X+Y=n)=sum_i=1^infty i p_x(X=i) + sum_j=1^infty j p_y (Y=j)=E[X]+E[Y]$$

The integral interpretation, also, doesn't help me in figuring this out. I wonder, also, if there is a criterion such that we know when integrating $phi$ is additive with respect to the random variables. For instance, the expected value is found taking $phi=x+y$, and we have additivity.

$$sum_nnmathsfPleft(X+Y=nright)=sum_nsum_i+j=nleft(i+jright)mathsfPleft(X=iwedge Y=jright)=$$$$sum isum_jmathsfPleft(X=iwedge Y=jright)+sum jsum_imathsfPleft(X=iwedge Y=jright)=$$$$sum imathsfPleft(X=iright)+sum jmathsfPleft(Y=jright)$$

The formula you have written is not correct. $phi$ is a function of two variables and you should write $int phi (x,y), dP_X,Y$ on the right side. $phi=1$ doesn't give you anything. It gives you $1=1$!. For $E(X+Y)=EX+EY$ take $phi (x,y)=x+y$ and use the fact that marginals of $P_X,Y$ are $P_X$ and $P_Y$.