Does there exist an absolutely continuous probability measure on every measure space?

Question

Let $(\Omega,\mathcal F,\mu)$ be an arbitrary measure space, where $\mu$ is non-zero but does not need to be $\sigma$-finite or semi-finite. Does there necessarily exist a probability measure $P$ on $(\Omega,\mathcal F)$ such that $P$ is absolutely continuous with respect to $\mu$?

Answer 1

If there are no $\mu-$finite sets, the answer is no. Otherwise, take any set $X\in \mathcal F$ s.t. $\mu(X)<\infty $. Then, define for all $A\in \mathcal F$, $$\mathbb P(A):=\frac{\mu(X\cap A)}{\mu(X)}=\int_A\frac{\boldsymbol 1_{X}(x)}{\mu(X)}\mu(\mathrm d x).$$ Obviously, $\mathbb P$ is a probability measure on $\Omega $ which is absolutely continuous w.r.t. $\mu$.