density function and measure null

Question

If $P = fd\mu $, $Q = gd\mu$ and $P = h dQ$ do we have $$ \mu(g=0)=0 $$ My goal is to show that $h = \frac{f}{g} $ almost $\mu$ everywhere.

thanks and regards.

Answer 1

Your notation is slightly wrong. Instead of $P = fd\mu$ you should write correctly $dP = fd\mu$, additionally we get then $dQ = gd\mu, dP = hdQ$

Simple plug in leads to:$$fd\mu = dP = hdQ = hgd\mu$$

So you get: $$f = hg$$ what's slightly different then $h = \frac{f}{g}$. But if $g \not= 0$ holds everywhere it's true. So in general you cannot conclude $$\mu(g = 0) = 0$$ You need some further assumptions for $P$ and $Q$ or $f$ and $h$ for it.