From the book **Billingsley - Convergence of probability measures, 1999,** we have the following definitions of tightness and relative compactness and the Prohorov's theorem:

**Tightness:** Let $\Pi$ be a family of probability measures on $(S,\mathcal{F})$. The family $\Pi$ is tight if for every $\epsilon$ there is a compact set $K$ such that $P(K)>1-\epsilon$ for every $P$ in $\Pi$. **Relative compactness:** We call $\Pi$ relatively compact if every sequence of elements of $\Pi$ contains a weakly convergent subsequence. **Prohorov's theorem:** Let $S$ be a Polish space and $\Pi$ a collection of probability measures on $S$. Than $\Pi$ is tight if and only if it is relatively compact.

Remark: in the literature on the evolution stochastic partial differential equations it is usually said that $\Pi$ is tight or relatively compact in $S$.

**My question:** *If I show tightness of family $\Pi$ in some set $A$, and $A\subset B$ (or $A\hookrightarrow B$ or $A\hookrightarrow_c B$, where $\hookrightarrow$ and $\hookrightarrow_c$ represent continuous and compact embeddings), is it true then that family $\Pi$ is tight in $B$ also?*

**In the concrete case:** I am working with family $\Pi$ that is a family of probability laws of solutions of some stochastic partial differential problem. And I am mostly interested in the next three cases:

$A=C([0,T];H^r(\mathbb{R}))$, $B=L^2(0,T;L^2(\mathbb{R}))$, where $H^r$ is the Sobolev space of $L^2$-type and $r\geq2$ is integer.

$A=C([0,T];H^r([-a,a]))$, $B=L^2(0,T;L^2(\mathbb{R}))$, where $[-a,a] \subset \mathbb{R}$ is a bounded interval.

$A=C([0,T];H^r([-a,a]))$, $B=C([0,T];H^r(\mathbb{R}))$.

**I think** that in the three cases mentioned the answer should be yes.

**My reasoning is:** If we talk about the relative compactness (for the type of spaces I mentioned above) and if $\Pi$ is relatively compact in $A$ and $A\subset B$, then $\Pi$ should be relatively compact in $B$ also. If that is true, the same conclusion is valid in the cases $A\hookrightarrow B$ or $A\hookrightarrow_c B$. By the Prohorov's theorem tightness is equivalent to relatively compactness. So if the family $\Pi$ is tight in $A$ it should be tight in $B$ if I am not mistaken?

If this is correct, it should save me some time in the problem I am working on (I won't need to do tightness then). Help with this would be great and I am $99\%$ sure that the answer is yes. But that $1\%$ chance is bugging me. Thanks in advance.