Let {$a_n$} be a sequence of non-negative sequence of real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. Show that $\int_Ef=\sum{a_n}$. The summation is infinite.
Here is my proof:
Consider the increasing sequence of non-negative measurable functions
${f}_{n}=\sum _{k=1}^{n}{a}_{k}{\chi }_{[k,k+1)}$ converging pointwise on $E$ to $f$. By Monotone Convergence Theorem we conclude that
$\int _{E}^{}f=\underset{n\to \infty }{\lim}\int _{E}^{}{f}_{n}=\underset{n\to \infty }{\lim}\sum _{k=1}^{n}\int _{E}^{}{a}_{k}{\chi }_{[k,k+1)}=\sum _{k=1}^{\infty }{a}_{k}$
Is that correct? If I am missing anything please let me know. Any help or idea will be appreciated.