Let $(E, A, \mu)$ be a measure space, $g: E \to [0, +\infty)$ be a positive measurable function.

Question

$v: \mathcal{A} \to [0, +\infty]$ by $v: A \mapsto \int_A g d\mu$

(a) Let $f: E \to [0, +\infty)$ be positive and measurable. Show that (*)$$\int_E f dv = \int_E g d\mu$$ (b) Show that f is $v$-integrable if and only if $fg$ is $\mu$-integrable, and that in this case (*) also holds.
In this question, we assume that $E$ is locally compact, Hausdorff, and that every open set of $E$ is $\sigma$-compact. We take $\mathcal{A}$ to be the Borel $\sigma$-algebra of $E$ and $\mu$ to be regular. Finally we assume that $g \in L^p(\mu)$ for some $1\leq p \leq \infty$. Show that $v$ is regular.

As a thought, would the Radon-Nikodym Theorem be useful for (1)? I think it might be unless that would be putting the cart before the horse — perpetuallyconfused, Apr 02 '20 at 03:11
You don't need the Radon-Nikodym theorem for this. It's an elementary measure theory result. — Alex Lapanowski, Apr 03 '20 at 05:41

score 0 · Answer 1 · answered Apr 03 '20 at 05:40

1 (a) should be $$ \int_{E}f\,d\nu = \int_{E} f\,g\,d\mu. $$

There's a routine for these measure theory type results which can be summarized as follows: prove the result for indicator functions, extend it to simple functions through linearity, and then extend it to measurable functions by monotone limits plus some limit theorem of your choice.

Why not try to prove 1 (a) for when f is an indicator function $f = \mathbf{1}_{A}$ of some measurable set $A$?

How would you go for the second part of the question? – Salamendrine Apr 03 '20 at 09:35 — Salamendrine, Apr 03 '20 at 09:35

Let $(E, A, \mu)$ be a measure space, $g: E \to [0, +\infty)$ be a positive measurable function.

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