A problem about two sigma-algebras and two respective measures.

Question

Apologies for the vague title, didn't know how to summarize the situation in a title-sized space. Suppose there are two measure spaces; $(X,S,\mu_1)\ \text{and}\ (X,T,\mu_2)$ such that the underlying set $X$ is the same, $S\subset T$ and for every $E\in S,\ \mu_1(E)=\mu_2(E)$. For some $S$-measurable function $f:X\to [0,\infty]$, can there exist a set $E^*\in X$ such that $$\mu_2(E^*)>0,\ \inf_{E^*}f>0,\ E^*\notin S,\ f(E^*)\notin B(\mathbb R_{\ge0}),$$ where $B(\mathbb R_{\ge0})$ denotes the Borel sets of the nonnegative reals? The conditions for $E^*$ would seem quite harsh, but i don't see a direction to look for a proof either way. If $\mu_1(X)$ would be finite, then one maybe might use Egorov's theorem to show that the size of the complement of $(X\setminus E)$ is very small for some $E\in S$, but in this situation it would need to work for a set $X$ with possibly infinite measure.