Let $f_k,f: \mathbb{R} \rightarrow \mathbb{R}$ for $k \in \mathbb{N} $ be integrable. I want to find a counterexample for the following statement:
$$ \lim_{n\to\infty} \int_\mathbb{R}f_n \text{d}\lambda = \int_\mathbb{R}f\text{d}\lambda \hspace{3mm}\Rightarrow \hspace{3mm} \lim_{n\to\infty} \int_\mathbb{R}|f_n-f|\text{d}\lambda = 0 $$
I know that the opposite implication holds, but I can't find a counterexample for this implication.
In general, we can not obtain some pointwise information from the information that the integrals are the same. One simple example is the following one.
Take $f_n = 1_{[n,n+1]}$ and $f =[0,1]$, then $\int_\mathbb{R} f_n \mathop{d \lambda} = \int f \mathop{d \lambda}$, but $\int_\mathbb{R} |f_n -f| \mathop{d \lambda} =2$ always. Of course, the same example works also for integration over $[0,1]$ instead of $\mathbb{R}$. Define $f_n = \frac{3}{(n+1)(n+2)} 1_{[(n+2)^{-1},(n+1)^{-1}]}$ and $f = 1_{[1/2,1]}$.