To show that a certain function is measurable.

Question

In preparation for the analysis qualifying exam, I am solving problems from various sources. Following is the problem I'm having difficulty with.

Suppose $(f_n)$ is a sequence of measurable, complex valued functions defined on a measure space $\Omega$. Show that the function $h:\Omega\to\mathbb{N} \cup \{\infty\}$ defined by

$$h(x)=\#\{n:f_n(x)=0\}$$

is measurable.

My thought was to verify the definition but no sigma algebras on either the domain or the co-domain are specified. I'm not sure where to begin. I would appreciate hints more than a complete solution. Thanks for your time.

Answer 1

Here is the attempt, thanks to Daniel Fischer's hint. Corrections are welcome.

Define $E_n=\{x:f_n(x)=0\}$. Then

$$h(x)=\sum_{n=1}^{\infty}\chi_{E_{n}}(x)$$

Since each $E_n$ is measurable, its respective characteristic function is also measurable and so is their partial sums. And finally, the limit of a measurable sequence (the sequence of partial sums in this case) is measurable. Thus $h$ is measurable.