We have the measure space (X,A,$\lambda$.) I have to construct a serie $(u_j)^\infty$ (from j=1) and a function $u\in\mathcal{L}^1(\lambda)$, so it's true that $u_j->u$ and $\int_R\ u d \lambda=5$ and $\int_R\ u_j d \lambda=3$ for all j.
I think that u probably could be as example the indicatorfunction u=1_[-2,3] because $\int_R\ 1[-2n,3n] d \lambda=\lambda([-2,3]=3+2=5$. But how can I construct the serie?
Let $u_n=5I_{(0,1)}- 2nI_{(0,\frac 1 n)}$ and $u=5I_{(0,1)}$. Then $u_n \to u$ at every point, $\int u_n=3$ for all $n$ and $\int u=5$.