Counterexample to $\int \left(\sum_\limits{k=1}^{\infty} f_k\right) d\mu = \sum_\limits{k=1}^{\infty} \int f_k d {\mu}$

Question

I search for a counterexample for the statement

$\int \left(\sum_\limits{k=1}^{\infty} f_k\right) d\mu = \sum_\limits{k=1}^{\infty} \int f_k d {\mu}$,
$f_k:X\to [-\infty, \infty]$ for all $k \in N$,
$\sum_\limits{k \in N}|f_k(x)|<\infty$ for all $x \in X$,
where $\mu $ is a measure.

Answer 1

My standard example is similar to the one in the comments, but uses rectangles instead of triangles. Define $g_k=k*I_{\left(0,\frac{1}{k}\right)}$, $f_1=g_1$, $f_n=g_n-g_{n-1}$ for $n>1$. The rest is the same.