In a book I saw that there are two conditions for a lemma:
1) $$\mathbb{E}\vert g(X_1,\ldots, X_m)\vert^r < \infty$$ and
2)$$\int \vert g(X_1,\ldots, X_m)\vert^r dF(X_1) \ldots dF(X_m) < \infty$$
here g is a permutation-symmetric function into the real numbers and with permutation symmetric I just mean: $$g(X_1, \ldots, X_i, \ldots, X_j, \ldots,X_M)=g(X_1, \ldots, X_j, \ldots, X_i, \ldots,X_M),$$ so we can just change the aruments among each other without changing the result;
But isn't (1) and (2) exactly the same? Isn't by definition given that? $$\mathbb{E}\vert g(X_1,\ldots, X_m)\vert^r=\int \vert g(X_1,\ldots, X_m)\vert^r dF(X_1) \ldots dF(X_m)$$?
Thanks for your help.
The two conditions are identical only if $F(X_1,\cdots,X_n)=F(X_1)\cdots F(X_n)$, namely, that $X_1,\cdots,X_n$ are independent. In other cases, the expectation and the integration are not the same.