Narrow convergence of measures and intermediate convergence of BV functions

Recall By the Riesz-Alexandroff representation theorem the dual of the space $C_0(\Omega,\mathbb{R}^N)$ (and thus of $C_c(\Omega,\mathbb{R}^N)$ can be isometrically identified with $M(\Omega,\mathbb{R}^N)$. i.e. any Borel measure $\mu$ is a bounded linear functional (continuous linear form if you like that) on $C_0(\Omega,\mathbb{R}^N)$ or $C_c(\Omega,\mathbb{R}^N)$. Thus, the weak convergence of a sequence of Borel measures $\mu_n$ is defined as follows.

Weak convergence of Borel measures: We say that a sequence of Borel measures $\mu_n$ converges weakly to $\mu$ i.e. $\mu_n\rightharpoonup \mu$ if $\langle\mu_n, \phi \rangle \rightarrow \langle \mu, \phi\rangle $ for all $\phi \in C_0(\Omega,\mathbb{R}^N)$.

Here, $\langle \mu, \,\phi\rangle\equiv\mu(\phi)\equiv\int_{\Omega} \phi \, d\mu \equiv \sum_{i=1}^N\int_{\Omega} \phi_i d\mu_i$.

Narrow convergence: A sequence $\{\mu_n\}$ in $M(\Omega, \mathbb{R}^N)$ narrowly converges to $\mu$ ($\mu_n\rightharpoonup \mu$ narrowly) in $M(\Omega, \mathbb{R}^N)$ if and only if $$\int_{\Omega} f\, d\mu_n \rightarrow \int_{\Omega} f d\mu,$$ for all $f \in C_b(\Omega, \mathbb{R}^N)$.

Narrow convergence is stronger than weak convergence.
If $\Omega=(0, 1)$ and $\mu_n=\delta_{\frac{1}{n}}$ then $\mu_n\rightharpoonup 0$ weakly. This is because for all $\phi \in C_0(\Omega)$, we have, $\mu_n(\phi)=\phi(\frac{1}{n})\rightarrow \phi(0)=0=\mu(\phi)$.

On the other hand, if $\phi\equiv1$ on $\Omega$, (note that $\phi \not\in C_0(\Omega)$ but $\phi \in C_b(\Omega)$), then we see that $\int_{\Omega} d\mu_n=\mu_n(\Omega)=1$ for all $n$. Essentially what happened with $\mu_n=\delta_{\frac{1}{n}}$ is that it converges weakly to $\mu=0$, but $\mu_n(\Omega)$, the size of $\Omega$ measured in $\mu_n$ does not converge to the size of $\Omega$ in measure $\mu(=0)$, that is $\mu_n\rightharpoonup \mu$ but $\mu_n(\Omega)\nrightarrow \mu(\Omega)$.

We have the following compactness result for the narrow topology.

PROKHOROV's sequential compactness theorem
If for a bounded subset $\mathcal{H}$ of $M^+(\Omega)$ it is true that for all $\epsilon$ there exists a compact subset $K_{\epsilon}\subset \Omega$ such that $\sup\{\mu(\Omega\ K_{\epsilon}):\mu\in \mathcal{H}\}\leq \epsilon$, then $\mathcal{H}$ is sequentially compact for the narrow topology.

The following equivalence proposition says that the convergence $\mu_n(\Omega)\rightarrow \mu(\Omega)$ is precisely what we need to ensure narrow convergence of measures.

Proposition
Let $\mu_n$ and $\mu$ be in $M^+(\Omega)$, non-negative Borel measures, then the following assertions are equivalent:
(i) $\mu_n\rightharpoonup \mu$ narrowly.
(ii)$\mu_n(\Omega)\rightarrow \mu(\Omega)$ and $\mu \rightharpoonup \mu$ weakly.

Proof
Note that as narrow convergence is stronger than weak convergence, it is clear that if $\mu_n\rightarrow \mu$ narrowly then $\mu \rightharpoonup \mu$. Also, $\mu_n(\Omega)\rightarrow \mu(\Omega)$ follows by taking $f\equiv 1$.

So, we need to only prove that if $\mu \rightharpoonup \mu$ weakly and $\mu_n(\Omega)\rightarrow \mu(\Omega)$ then $\mu_n\rightharpoonup \mu$ narrowly. That is, we need to show that for any function $f$ in $C_b(\Omega, \mathbb{R}^N)$ we have $\big| \int_{\Omega} fd\mu_n - \int_{\Omega} fd\mu \big|\rightarrow 0$. To do this we add and subtract $\int_{\Omega} f\,\phi d\mu_n$ and $\int_{\Omega} f\,\phi d\mu$ where $\phi$ is continuous function that is compactly supported in $\Omega$. The result follows as $f$ is bounded i.e. $\Vert f \Vert_{\infty}$ is a finite number and $\int_{\Omega} f\phi d\mu_n \rightarrow \int_{\Omega} f\phi d\mu$ due to compact support of $\phi$ in $\Omega . \hspace{25pt}\square$

More equivalence proposition
Let $\mu_n$ and $\mu$ be in $M^+(\Omega)$, non-negative Borel measures, then the following assertions are equivalent:
(i) $\mu_n\rightharpoonup \mu$ narrowly.
(ii) $\mu_n(\Omega)\rightarrow \mu(\Omega)$ and $\mu(U)\leq \lim\inf_{n\rightarrow \infty} \mu_n(U)$ for all open subsets $U\subset \Omega$.
(iii) $\mu_n(\Omega)\rightarrow \mu(\Omega)$ and $\mu(F)\geq \lim\inf_{n\rightarrow \infty} \mu_n(F)$ for all closed subsets $F\subset \Omega$.
(iv) $\mu_n(B)\rightarrow \mu(B)$ for all Borel subsets $B\subset\Omega.\,\,\hspace{25pt}\square$

Ok, so far so good. But images are not necessarily continuous functions. They might have jump discontinuities, i.e. edges. For this the following result due to Marle (Measure et probabilities) is helpful:

Proposition
Let $\mu_n$ and $\mu$ be non-negative Borel measures $M^+(\Omega)$, with $\mu_n\rightharpoonup \mu$ narrowly and let $f$ be a $\mu_n$ measurable bounded function for all $n$, such that it is discontinuous only on a set of null $\mu-$ measurable set then
$$\hspace{25pt}\int_{\Omega} f\, d\mu_n \rightarrow \int_{\Omega} f\, d\mu \hspace{25pt}\square$$

Now, that we have seen a notion of convergence for sequence of measures that is stronger than weak convergence, let's look at the a notion of convergence for sequence of functions in $BV$ space that is stronger than weak convergence.

Intermediate convergence
Let $\{u_n\}$ be a sequence in $BV(\Omega)$ and $u\in BV(\Omega)$. we say that $u_n\rightharpoonup u$ intermediately (or in the sense of intermediate convergence) if and only if
$$u_n\rightarrow u \mbox{ in } L^1(\Omega),$$
$$\hspace{25pt}\vert u_n\vert_{BV}\rightarrow \vert u\vert_{BV}\,\,\hspace{25pt}\square$$

Recall, from the previous post that for $\{u_n\}\in BV(\Omega)$ with bounded TV mass and $u_n\rightarrow u$ strongly in $L^1(\Omega)$, then we have $u_n\rightharpoonup u$ in $BV$, but $\vert u \vert_{BV}\leq \lim\inf_{n\rightarrow \infty}\vert u_n \vert_{BV}$ i.e. some of the mass is lost.

Ok, now I am hungry, so more about this tomorrow.