Relaxing hypotheses of Master Theorem
Yes. Here is the idea.

If $f(n) = \Theta(g(n))$ then there exist constants $C_1,C_2&gt;0$ such that for large $n$,
$$ C_1g(n) \leq f(n) \leq C_2g(n). $$
Now consider the two recurrences
$$
T_1(n) = aT_1(n/b) + C_1g(n) \\
T_2(n) = aT_2(n/b) + C_2g(n)
$$
Choosing suitable initial conditions, you can prove inductively that
$$
T_1(n) \leq T(n) \leq T_2(n).
$$
You consider a regime in which applying the master theorem gives you that $T_1(n) = \Theta(g(n))$ and $T_2(n) = \Theta(g(n))$. It follows that $T(n) = \Theta(g(n))$.