When $f:\mathbb N \to \mathbb R$ is $\lambda$ integrable?

Question

I'm studying measure theory for tomorrow's exam and this question came up. Any help would be great!

My question is

• When is $f:\mathbb N \to \mathbb R$ $\lambda$-integrable?

• And if it is what is the integral of $f$?

Answer 1

Let $\lambda$ be the counting measure on $\mathbb N$. Suppose that $f(n)=a_n$, for all $n\in\mathbb N$. Any subset of $\mathbb N$ with the counting measure is measurable, so $f$ is measurable. Now, you have that $$\int_{\mathbb N}|f(n)|\,d\lambda(n)=\int_{\mathbb N}\sum_{k\in\mathbb N}\chi_k(n)|f(n)|\,d\lambda(n)=\sum_{k\in\mathbb N}\int_{\{k\}}|f(n)|\,d\lambda(n)=\sum_{k\in\mathbb N}|f(k)|.$$ So, $f$ is integrable when the series of $(a_n)$ is asbolutely convergent. In this case, calculating as above, we have that $$\int_{\mathbb N}f(n)\,d\lambda(n)=\sum_{k\in\mathbb N}\int_{\{k\}}f(n)\,d\lambda(n)=\sum_{k\in\mathbb N}f(k)=\sum_{k\in\mathbb N}a_k.$$

Answer 2

Let $\mu(A) =\sum_{n=1}^{\infty}a_n\delta_n(A)$ where $a_n=f$ and $\delta_n$ the Dirac measure.So $\mu$ is finite iff $\sum_{n=1}^{\infty}a_n$ is convergeant.