I've been learning Lebesgue Integral recently and the following is a problem in one of our previous sample exams:
If {$f_n$} is a bounded sequence of functions in $L^1(\mathbb{R})$, $f_n\to f$ a.e. and
suppose that for any $M>0$, there exists $N\in\mathbb{N}$ such that $$\int_{\mathbb{R}\backslash[-M,M]}|f_N|>1$$ Prove that $||f_n||_1$ does NOT converge to $||f||_1$.
I tried to separate the integral $||f_n||_1$ into two parts: $[-M,M]$ and $\mathbb{R}\backslash[-M,M]$. The first part converges due to the Bounded Convergence Theorem. So it suffices to prove the second part does NOT converge. But I don't know where to go from there.
I also thought of functions, e.g. $\chi_{[-n-1,-n]\cup[n,n+1]}$ that satisfy the given conditions but had no idea how to use the properties of those functions. Any hint will be much appreciated!
Best Answer