Why does pointwise convergence of a function not imply convergence of the integral of that function?

Question

The question asks: If $f_n\to f$ pointwise, then $\int f_n \to \int f$, where I have to provide a proof or a counter example. I know that it is not true, and hence I must provide a counter example, but how might I go about doing this?

Answer 1

Let $f_n(x)= \frac 1x$ for $1 \leq x \leq n+1, - \frac {1}{x-n}$ for $n+1 \lt x \leq 2n+1$, and $0$ for $x \lt 1$ and $x \gt 2n+1$.

Then $\int f_n = 0$ but $f_n \to \frac 1x$ for $x \geq 1, 0$ for $x \lt 1$ pointwise.

Answer 2

Consider $$f_n(x)=\max\{0,2n-|n^2x-n|\} $$ on $[0,2]$.