I am trying to prove the following theorem, which I feel should be true, but am not sure how to go about it.
Suppose we are in a probability space $(\Omega, \mathcal{F}, P)$ and we define for any (possibly non-measurable) subset:
$$P^*(A) = inf \{P(B): A \subset B, B \in \mathcal{F}\}$$
Now, I want to show $P^*(M) = 1$ iff $P(A) = 0$ for all $A \subset \Omega\setminus M$ where $A \in \mathcal{F}$. Is this true? If so, how is it proven?
Yes, it is true. Suppose that $P^*(M)$ is not 1, but it is 0.9, say. Then there is some $B\in\mathcal{F}$ such that $M\subset B$ and $P(B)$ is arbitrarily close to 0.9, for instance less than 0.95. Then $A=\Omega-B$ has measure at least 0.05, which is absurd.
Conversely, suppose that $P^*(M)=1$ and that there is some $A\in\mathcal{F}$ which does not meet $M$ and such that $P(A)>0$, for instance $P(A)=0.05$. Then $B=\Omega-A\in\mathcal{F}$ and $P(B)=1-0.05=0.95$, but this is absurd because $M\subset B$ and the infimum is less than 0.95.