Subsequence of $L^{2}(\Omega)$ - bounded sequence weakly * converging to a measure

Question

I was reading a well available article in the internet: "THE COMPENSATED COMPACTNESS METHOD APPLIED TO SYSTEMS OF CONSERVATION LAWS by Tartar"; there at Page-266, it is written:

"$L^{1}(\Omega)$ is isometrically imbedded into $\mathcal M_{b}(\Omega)$ (-the space of measures with finite total mass) which is the dual of $C_{b}(\Omega)$ (-the space of continuous bounded functions with Sup. norm).

[This much I know ... so for me this is clear... now, in the next line it is written that:]

From a bounded sequence: $f_{n}$ in $L^{2}(\Omega)$ one can extract a subsequence $f_{n_{m}}$ weakly * converging to a measure $\mu$ i.e. $\int_{\Omega} f_{n_{m}}g dx \rightarrow <\mu,g> \forall g \in C_{b}(\Omega)$ ."

Now, I am a bit clueless about this second line. Can someone give me a proof of it??

Thank You!!

Answer 1

If $\{f_n\}$ is bounded in $L^2(\Omega)$, it is also bounded in $L^1(\Omega)$ and, hence, in $\mathcal{M}_b(\Omega)$. Now, $\mathcal{M}_b(\Omega)$ is the dual space of the seperable Banach space $C_b(\Omega)$. Hence, $\{f_n\} \subset \mathcal{M}_b(\Omega)$ has a weak-* convergent subsequence.