A consequence of Fubini's theorem for non-negative functions is that for any $p>0$,
$$\int_{\Omega}g(x)^pdx=\int_0^{+\infty}p\mu_g(t)t^{p-1}dt=\sum_{k=0}^{+\infty}\int_{\eta M^k}^{\eta M^{k+1}}p\mu_g(t)t^{p-1}dt.$$
Let $a_k:=\int_{\eta M^k}^{\eta M^{k+1}}p\mu_g(t)t^{p-1}dt$. Consider the case $p\geq 1$. Since $\mu_g$ is decreasing and $t\mapsto t^{p-1}$ is increasing, we have
$$\mu_g(\eta M^{k+1})\eta M^k(M-1)\eta^p (M^k)^{p-1}\leq a_k\leq \mu_g(\eta M^k)(M-1)M^k\eta^{p-1}(M^{k+1})^{p-1},$$
hence
$$C_1\mu_g(\eta M^{k+1})M^{(k+1)p}\leq a_k\leq C_2\mu_g(\eta M^k)M^{kp}.$$
This gives the wanted equivalence.
When $p<1$, a similar argument applies.