Existence of a function in Radon Measure

Question

I am having trouble to construct a function $f$ in the following problem.

Let $X$ be Locally Compact Hausdorff topological space. Let $\mu$ is a positive Radon measure on $X$ with $\mu(X)=\infty$ then there exists $f\in{C_{0}(X)}$ such that $\int_{X}fd\mu=\infty$.

Any ideas?

Thanks!

Answer 1

Hint. First construct a sequence of compact sets $K_n\subset\subset K_{n+1}$, such that $$\lim_{n\to\infty}\mu(K_n)=\infty.$$

Define $f_n\ge 0$ continuous to but equal to $1$ in $K_n$ and $0$ in $K_{n+1}^c$.

Then look for $f=\sum a_nf_n$, for suitable $a_n>0$, $\sum a_n<\infty$, so that $\int_X f\,d\mu=\infty$ and $f\in C_0$.