$L^p(X,\mu)$, say $\{f_k\}_{k \in \{1,...,N\}}$, where $C$ is the constant from Proposition \[wlozenie\]. Now it is enough to prove that $\{f_k\}_{k \in \{1,...,N\}}$ is an $\epsilon$ net of $\mathfrak{B}$ in $L^q(X,\mu)$; indeed, by the interpolation inequality we have $$\begin{aligned}
        \|f-f_k\|_{L^q(X,\mu)}&\leq&\|f-f_k\|^{\theta}_{L^p(X,\mu)}\|f-f_k\|^{1-\theta}_{L^{p^*}(X,\mu)}\\
        &\leq&C^{1-\theta}\|f-f_k\|^{\theta}_{L^p(X,\mu)}\|f-f_k\|^{1-\theta}_{M^p_1(X,\mu)}\\
        &\leq&2^{1-\theta}C^{1-\theta}\|f-f_k\|^{\theta}_{L^p(X,\mu)}\leq \epsilon\end{aligned}$$ for some $k$ in $\{1,...,N\}$.

Proposition \[noncritic\] highlights the fact that in general one cannot expect that $M^p_1(X,\mu)  \hookrightarrow \hookrightarrow L^{p^*}(X,\mu)$. Theorem \[main2\] ensures that some proper subset of $M^p_1(X,\mu)$ is relatively compact in $L^{p^*}(X,\mu)$.

Auxiliary Lemmata
-----------------

The next lemma seems to be well known, however we give a detailed proof due to its role in Section \[main\].

\[lemma:delty\] Here $(\Omega ,d , \mu)$ is a separable metric-measure space with a finite Borel measure $\mu$. Suppose that there exists some $\delta >0$ such that for every measurable set $A$, either $\mu(A) = 0$ or $\mu(A) \geq \delta$. Then there exists a finite set $\{x_i \}_{i \in I}$ of points in $\Omega$ and a finite set of numbers $\{ \mu_i \}_{i \in I}$ not smaller than $\delta$ such that $$\mu = \sum_{i \in I} \mu_i \delta_{x_i}.$$

Consider the set $A_{\delta} := \left\{ x \in \Omega \, : \, \mu(x) \geq \delta \right\}$. Since $\mu(\Omega) < \infty$, the 