Royden's Real Analysis Question: Let {$a_n$} be a sequence of nonnegative real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. I want to show that $\int_Ef=\sum{a_n}$. The summation is infinite.
I don't even know how to approach this problem. It's just so confusing. Any help will be greatly appreciated.
Thankyou!
$$\int _E f=\int_{\bigcup\limits_{n=1}^\infty[n,n+1)}f=\int_Ef\cdot\chi_{\bigcup\limits_{n=1}^\infty[n,n+1)}=$$ $$\int_Ef\cdot \sum\limits_{n=1}^\infty\chi_{[n,n+1)}=\int_E\sum\limits_{n=1}^\infty f\cdot\chi_{[n,n+1)}\overset{\mbox{Levi}}{=}$$ $$\sum\limits_{n=1}^\infty\int_Ef\cdot\chi_{[n,n+1)}=\sum\limits_{n=1}^\infty \int_{[n,n+1)}f=$$ $$\sum\limits_{n=1}^\infty \int_{[n,n+1)}a_n=\sum\limits_{n=1}^\infty a_n\cdot\mu ([n.n+1))=$$ $$\sum\limits_{n=1}^\infty a_n$$
Beppo Levi's theorem is an application of Monotone Convergence theorem which allows the switch between integral and infinite sums of non-negative terms.