Complex Measures: Pushforward

Question

Problem

Given Borel spaces $X$ and $Y$.

Consider a complex measure: $$\mu:\mathcal{B}(X)\to\mathbb{C}$$

Regard a pushforward: $$h\in\mathcal{B}(X,Y):\quad\nu:=\mu\circ h^{-1}$$

Then one has: $$|\nu|=|\mu\circ h^{-1}|=|\mu|\circ h^{-1}$$

How can I check this?

Reference

This is a note for: Pushforward (SM)

Answer 1

I don't think this is correct.

Let us consider $X =\{\pm1\}$ and $Y =\{0\}$ as well as $h \equiv 0$ and $\mu =\delta_1 - \delta_{-1}$.

It is not hard to see $\nu \equiv 0$ and hence $|\nu|\equiv 0$ but $|\mu|=\delta_1 +\delta_{-1}$.

That implies:

$$ |\nu|(Y)=0\neq2=|\mu|(X)=|\mu|(h^{-1}(Y)). $$