Suppose that $(f_n)\in M^+(X, X)$ and $(f_n)$ converges to $f$, and that $$\int f \, d\mu=\lim \int f_n \, d\mu\lt +\infty,$$ we have the conclusion that $$\int_E f \, d\mu=\lim \int_E f_n \, d\mu$$ for each $E\in X$
Note that $M^+(X, X)$ denote the collection of all extended real-valued $X$-measuable functions on $X$
I already know how to prove the above claim, but could someone show that the conclusion may fail if the condition: $\lim \int f_n \, d\mu\lt +\infty$ is dropped. Thanks.
Define $$ f_n = \chi_{[-n,0]} + \chi_{[n,n+1]},f = \chi_{]-\infty,0]} $$ then $f_n \to f$ pointwise, and $\int f\,d\lambda = \lim \int f_n\,d\lambda$. (Here $\lambda$ is Lebesgue measure on $\mathbb{R}$).
But if you choose $E = [0,\infty[$...