Let $<\Omega,\mathfrak{F},\mathbb{P}>$ be a probability space and let $\mathfrak{F}_1\cup\mathfrak{F}_2 =\mathfrak{F}$ be independent $\sigma$-fields.
Then do there exist singular measures $\mathbb{P}_1$ and $\mathbb{P}_2$ such that
$$
\mathbb{P}=\mathbb{P}_1+\mathbb{P}_2
$$
and
for every $f$ and $g$ which are $\mathfrak{F}_1$ and $\mathfrak{F}_2$ measurable respectively.
$$
\int f d\mathbb{P}_2 =0 \, \ \ \text{and} \ \ \,\int g d\mathbb{P}_1 =0?
$$