The coarea formula for BV functions

Today, we will discuss the coarea formula for $BV$ functions, established by Fleming and Rishel in 1960 (W. Fleming, R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218-222.)

Let us first recall the lower semicontinuity property of the total variation with respect to the strong convergence in $L^1(\Omega)$:
Let $\{u_n\}_{n\in\Omega}$ be a sequence in $BV(\Omega)$ strongly converging to some $u$ in $L^1(\Omega)$ and satisfying $|u_{n}|_{BV} < \infty$. Then $u\in BV(\Omega)$ and $$|u|_{BV(\Omega)} \leq \lim\inf_{n\rightarrow \infty} |u_n|_{BV(\Omega)}.$$ The coarea formula is as stated as follows:

Coarea formula: Let $u$ be a given function in $BV(\Omega)$. Then for almost every $t$ in $\mathbb{R}$, the level set $E_t=\{ x\in \Omega \subset \mathbb{R}^N: u(x) > t\}$ of $u$ is a set of finite perimeter in $\Omega$, and
\begin{align}
&Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt,\\
&|u|_{BV(\Omega)}=|Du|(\Omega)=\int_{-\infty}^{\infty} \int_{\Omega}|D\chi_{E_t}|\, dt.
\end{align}
The second assertion above is often written in terms of the perimeter of level sets as follows:
\begin{align}
|u|_{BV(\Omega)}=\int_{-\infty}^{\infty} Per(E_t, \Omega) dt,
\end{align}
where $Per(E_t, \Omega)=\mathcal{H}^{N-1}(\Omega \cap \partial E_t)$ is the perimeter of the level set $E_t$.

Proof of the coarea formula: We follow the treatment of Variational Analysis in Sobolev and BV spaces by Attouch et al.

Part I: Proof of $|Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dt$:

Let us assume that $D\chi_{E_t}$ belongs to $\mathbf{M}(\Omega, \mathbb{R}^N)$, the set of all $\mathbb{R}^N-$ valued Borel measures which is the set of all the $\sigma-$ additive set functions $\mu:\mathcal{B}(\Omega)\rightarrow \mathbb{R}^N$, with $\mu(\emptyset)=0$. For all $t$ in $\mathbb{R}$ set
\begin{equation}
f_t=\left\{
\begin{array}{ll}
\chi_{E_t} & \mbox{if } t \geq 0 \\
-\chi_{\Omega \backslash E_t} & \mbox{if } t < 0 \end{array} \right. \end{equation} It is easy to see that $u(x)=\int_{-\infty}^{\infty}f_t(x)\,dt$ for all $x\in \Omega$. For all $\bar{\phi} \in C_c^1(\Omega, \mathbb{R}^N)$ we have \begin{align*} &\langle Du, \bar{\phi} \rangle \\ &= -\int_{\Omega} u \, \mbox{div} \,\bar{\phi} \, dx \\ & = - \int_{\Omega} \int_{-\infty}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx\\ &= - \int_{\Omega} \int_{-\infty}^{0} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \chi_{\Omega \backslash E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} (1-\chi_{E_t}) \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx - \int_{\Omega} \int_{-\infty}^{0} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= 0-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{-\infty}^{\infty} \langle D\chi_{E_t}, \bar{\phi}\rangle \,dt \end{align*} Hence, $Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt$, and $|Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dx dt$.

Part II: (coverse) Proof of $\int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dxdt\leq |Du|(\Omega) $:


Step i. We first assume that $u$ belongs to the space $\mathcal{A}(\Omega)$ of piecewise linear and continuous functions in $\Omega$. By linearity, one can assume that $u=a\cdot x+b$ with $a\in \mathbb{R}^N$ and $b\in \mathbb{R}$, so that

\begin{align*}
&\int_{\Omega} |D\chi_{E_t}|\\
&=\mathcal{H}(\Omega \cap \partial E_t)\\
&=\mathcal{H}(\Omega \cap \{x| a\cdot x +b=t\})\\
&=\int_{\Omega \cap \{x| a\cdot x +b=t\}} d\mathcal{H}^{N-1}(x)\\
&=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x)
\end{align*}
Thus we get,
\begin{align*}
&\int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\\
&=|a|\mathcal{L}(\Omega)\\
&=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x)\\
&=\int_{\Omega} |Du|
\end{align*}
Hence we have,
\begin{align*}
\int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt=|Du|(\Omega) \dots \mbox{ if } u=a\cdot x+b.
\end{align*}

Step ii. Now we prove that $\int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\leq|Du|(\Omega)$ for all $u\in BV(\Omega)$. We know that the space of piecewise linear and continuous functions $\mathcal{A}(\Omega)$ is dense in $W^{1, 1}(\Omega)$ when equipped with the strong topology. We also know that the space $C^{\infty}\cap W^{1, 1}(\Omega)$ is dense in $BV$ when equipped with the intermediate convergence. Thus, there exists a sequence $\{u_n\}_{n\in \mathbb{N}} \in \mathcal{A}(\Omega)$ such that $u_n\rightharpoonup u$ for the intermediate convergence. For each of the functions $u_n$ we set $E_{n, t}:=\{x\in \Omega: u_n(x) > t\}$. Due to the intermediate convergence, we have
\begin{align*}
&\int_{\Omega}|Du|=\lim_{n\rightarrow \infty}\int_{\Omega}|Du_n| \dots \mbox{the intermediate convergence}\\
&=\lim_{n\rightarrow \infty} \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{from step i}\\
&\geq \int_{-\infty}^{\infty} \lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{Fatou's lemma}\\
\end{align*}
From the intermediate convergence we also have $u_n\rightarrow u$ in $L^1(\Omega)$. i.e.
\begin{align*}
\int_{\Omega} |u_n-u|=\int_{\Omega} \int_{-\infty}^{\infty} |\chi_{E_{n,t}}-\chi_{E_t}|\ dt dx = \int_{-\infty}^{\infty} \Big(\int_{\Omega} |\chi_{E_{n,t}}-\chi_{E_t}|\ dx \Big) dt=0.
\end{align*}
Thus, for a subsequence $E_{n_m, t}$, and for almost all $t \in \mathbb{R}$, $\chi_{E_{n_m, t}}\rightarrow \chi_{E_{n, t}}$ strongly in $L^1(\Omega)$. The lower semicontinuity of the total variation with respect to the strong convergence in $L^1(\Omega)$ we get $\lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n_m,t}}|\, dx \geq \int_{\Omega} |D\chi_{E_{t}}|\, dx$. Thus, we get,
$$ \int_{\Omega}|Du| \geq \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{t}}|\, dx dt. $$
This completes the proof!

(By the way, step ii also proves that $D\chi_{E_t} \in \mathbf{M}(\Omega, \mathbb{R}^N)$ for a.e. $t\in \mathbb{R}$, something that we used in Part I of the proof.)

More on intermediate convergence

According to the vectorial version of the Riesz-Alexandroff representation theorem, the dual norm $\vert u \vert_{BV}=\Vert Du \Vert=\sup \{\int_{\Omega} u \,\mbox{ div } \mathbf{\phi}: \mathbf{\phi} \in C_c^1(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1\}$ is also the total mass $|Du|(\Omega)=\int_{\Omega}|Du|$ of the total variation $|Du|$ of the measure $Du$.

Moreover, from classical integration theory, the integral $\int_{\Omega}\phi \,d(Du)$ is well defined for all $Du$ integrable functions $\phi$ from $\Omega$ into $\mathbb{R}^N$, e.g. for functions in $C_b(\Omega, \mathbb{R}^N)$. For the same reasons, $\int_{\Omega}\phi\, d|Du|$ is well defined for all $|Du|$ integrable functions $\phi$, in particular for functions in $C_b(\Omega)$.

Thus, we can say that $|Du_n|\rightharpoonup |Du|$ narrowly in $M(\Omega)$ if $\int_{\Omega}\phi \, d|Du_n|\rightarrow \int_{\Omega}\phi\,d|Du|$, for all $\phi \in C_b(\Omega)$. If we choose $\phi\equiv 1$ on $\Omega$ then $\Vert Du_n \Vert \rightarrow \Vert Du \Vert$ i.e. $|u_n|_{BV}\rightarrow |u|_{BV}$.

Hence, $u_n\rightarrow u$ in $L^1(\Omega)$ and the narrow convergence of measures, $|Du_n|\rightharpoonup |Du|$ in $M(\Omega)$ implies the intermediate convergence, $u_n\rightharpoonup u$ (i.e. $u_n\rightarrow u$ in $L^1(\Omega)$ and $|u_n|_{BV}\rightarrow |u|_{BV}$).


We have the following proposition regarding the intermediate convergence.

Proposition
The following three assertions are equivalent
(i) $u_n\rightharpoonup u$ in the sense of intermediate convergence.
(ii) $u_n\rightharpoonup u$ in $BV(\Omega)$ and $\vert u_n \vert_{BV}\rightarrow \vert u \vert_{BV}$.
(iii) $u_n\rightarrow u$ strongly in $L^1(\Omega)$ and $\vert Du_n \vert\rightharpoonup \vert Du \vert$ narrowly in $M(\Omega)$.

Proof
The equivalence (i) to (ii) is a consequence of a proposition from the previous post. We have just seen above that (i) and (iii) are equivalent. We only need to prove that (ii) implies (iii).

Recall that $u_n\rightharpoonup u$ weakly in $BV(\Omega)$ means that
(a) $u_n\rightarrow u$ strongly in $L^1(\Omega)$ and
(b) the measures $Du_n\rightharpoonup Du$ weakly in $M(\Omega, \mathbb{R}^N)$.

Also, recall that for non-negative Borel measures $\mu_n$ and $\mu$ in $M(\Omega)$ the following statements are equivalent:
(c) $\mu_n(\Omega)\rightarrow \mu(\Omega)$ and $\mu(U)\leq \lim\inf_{n\rightarrow\infty}\mu_n(U)$ for all open subsets $U\subset \Omega$.
(d) $\mu_n\rightharpoonup \mu$ narrowly in $M(\Omega)$.

Let, $\mu_n=|Du_n|$ and $\mu=|Du|$. We have been given that $|Du_n|(\Omega)\rightarrow |Du|(\Omega)$ i.e. we have $\mu_n(\Omega)\rightarrow \mu(\Omega)$. For showing the narrow convergence $\mu_n\rightharpoonup \mu$ we just need to prove the lower-semicontinuity property on $U\subset\Omega$,
i.e. we need to show that for $\mu(U)\leq \lim\inf_{n\rightarrow\infty}\mu_n(U)$ for all open subsets $U\subset \Omega$,
i.e. we need to show that for $|Du|(U)\leq \lim\inf_{n\rightarrow\infty}|Du_n|(U)$ for all open subsets $U\subset \Omega$.

To this effect we use the proposition from the previous blog which says that if $u_n\in BV(\Omega)$ with $|u_n|_{BV(\Omega)}$ bounded, with $u_n\rightarrow u$ in $L^1(\Omega)$, then we have the lower semicontinuity property i.e. $|Du|(\Omega)\leq \lim\inf_{n\rightarrow\infty}|Du_n|(\Omega)$, and $u\in BV(\Omega)$.

Let $U\subset\Omega$ be any open subset. We have $u_n\in BV(\Omega)$, thus as $U\subset\Omega$, we have $u_n\in BV(U)$. Thus, we have $\sup_n |Du_n|(U)\leq \sup_n |Du_n|(\Omega)<\infty$.

The weak convergence $u_n\rightharpoonup u$ in $L^1(\Omega)$ implies the strong convergence $u_n\rightarrow u$ in $L^1(\Omega)$ and thus the strong convergence $u_n\rightarrow u$ in $L^1(U)$ follows as $U\subset \Omega$.

 Now that we have the conditions in the proposition: $u_n\in BV(U)$, $\sup_n |Du_n|(U)<\infty$, and $u_n\rightarrow u$ in $L^1(U)$ we have the lower semicontinuity property: $|Du|(U)\leq \lim\inf_{n\rightarrow\infty}|Du_n|(U)$ for all open subsets $U\subset \Omega$. 

As we have $|Du_n|(\Omega)\rightarrow |Du|(\Omega)$ and $|Du|(U)\leq \lim\inf_{n\rightarrow\infty}|Du_n|(U)$ for all open subsets $U\subset \Omega$, from the equivalence of (c) and (d) we get the narrow convergence: $|Du_n|\rightharpoonup |Du_n|$ narrowly in $M(\Omega). \,\,\hspace{25pt} \square$.

In conclusion, intermediate convergence implies narrow convergence of $|Du_n|$.

There is something about the $BV$ space

I have been putting it off for a while. But now that I can write Latex in the blogger I can talk about $BV$ spaces.
Solutions of some mathematical problems which have discontinuities along one-codimensional manifolds where the first distributional derivatives are measures prompt us to consider the $BV$ space. I will refer to Attouch et al's book: Variational Analysis in Sobolev and BV spaces.

Definition of $BV$ space: We say that a function $u:\Omega\rightarrow \mathbb{R}$ is a function of bounded variations i.e. $u\in BV \,$ if and only if it belongs to $L^1(\Omega)$ and its gradient $Du$ in the distributional sense is in $M(\Omega, \mathbb{R}^N)$.

The following statements are equivalent.
(i) $u\in BV(\Omega)$
(ii) $u \in L^1(\Omega)$ and $u_{x_i}\in M(\Omega)$ for all $i=1, 2, ..., N$.
(iii) $u \in L^1(\Omega)$ and $\vert u \vert_{BV}<\infty$, where $\vert u\vert_{BV}:=\sup \{\langle Du, \,\mathbf{\phi}\rangle: \mathbf{\phi} \in C_c(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1\}.$
(here $\langle Du, \,\mathbf{\phi}\rangle:=\sum_{i=1}^N \int_{\Omega}\, \mathbf{\phi}_i u_{x_i}$.)
(iv) $u \in L^1(\Omega)$ and $\vert u \vert_{BV}=\sup \{\int_{\Omega} u \,\mbox{ div } \mathbf{\phi}: \mathbf{\phi} \in C_c^1(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1\}<\infty$.

The implication (ii)$\implies $(iii) follows as $C_c(\Omega, \mathbb{R}^N)$ is dense in $C_0(\Omega, \mathbb{R}^N)$. The implication (ii)$\implies $(iii) follows as $C_c^{\infty}(\Omega, \mathbb{R}^N)$ is dense in $C_0(\Omega, \mathbb{R}^N)$ and $C_c(\Omega, \mathbb{R}^N)$.

Notations and facts:
$\Omega \subset \mathbb{R}^N$.
$C_0(\Omega, \mathbb{R}^N)$ is the space of all continuous functions that vanish at infinity. i.e. for a function $\phi\in C_0(\Omega, \mathbb{R}^N)$ the following is true: Given any $\epsilon > 0 $ there exists a compact set $K_{\epsilon} \subset \Omega$ such that $\sup_{x\in\Omega \backslash K_{\epsilon}}|\phi(x)|\leq \epsilon$.
The functions $\mathbf{\phi}$ in $C_0(\Omega, \mathbb{R}^N)$ are equipped with the uniform norm
$\Vert \phi \Vert_{\infty}:= \sup_{x \in \Omega}\{ |\phi(x)|\}$.
$C_c(\Omega, \mathbb{R}^N)$ is the subspace of $C_0(\Omega, \mathbb{R}^N)$ with a compact support in $\Omega$.
$C_b(\Omega,\mathbb{R}^N)$ is the subset of all bounded continuous functions from $\Omega$ into $\mathbb{R}^N$.

Density results:
$C_c(\Omega, \mathbb{R}^N)$ is dense in $C_0(\Omega, \mathbb{R}^N)$.
The space of infinitely differentiable and compactly supported functions $C_c^{\infty}(\Omega, \mathbb{R}^N)$ is dense is $C_0(\Omega, \mathbb{R}^N)$ and $C_c(\Omega, \mathbb{R}^N)$.

Duality results:
$M(\Omega, \mathbb{R}^N)$ denotes the space of all $\mathbb{R}^N$ valued Borel measures.
$M(\Omega, \mathbb{R}^N)$ is isomorphic to the product space $M^N(\Omega)$.
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 $C_c(\Omega, \mathbb{R}^N)$ and the dual norms $\Vert\cdot \Vert_{C_{0}^{'}(\Omega, \mathbb{R}^N)}$ and $\Vert \cdot \Vert_{C_{c}^{'}(\Omega, \mathbb{R}^N)}$ are equal to the total mass $|\cdot|(\Omega)$ :
$$|\mu|(\Omega)\equiv \int_{\Omega}|\mu|=\sup\{ \langle \mu, \,\mathbf{\phi}\rangle: \mathbf{\phi} \in C_0(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1 \} =\sup\{ \langle \mu, \,\mathbf{\phi}\rangle: \mathbf{\phi} \in C_c(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1 \}.$$ Here, $\langle \mu, \,\phi\rangle\equiv\mu(\phi)\equiv\int_{\Omega} \phi \, d\mu \equiv \sum_{i=1}^N\int_{\Omega} \phi_i d\mu_i.$

The relation between $BV$ and the Sobolev space $W^{1, 1}$:

According to Radon-Nykodym theorem there exists $\nabla u \in L^1(\Omega, \mathbb{R}^N)$ and measure $D_s u$ that is singular with respect to $\mathcal{L}^N|_{\Omega}$, the $N-$ dimensional Lebesgue measure restricted to $\Omega$, such that: $$Du=\nabla u \, \mathcal{L}^N|_{\Omega}+ D_s u.$$ Thus, $W^{1, 1}$ is a subspace of the $BV-$ space and for functions in $W^{1, 1}$ we can write $Du=\nabla u$.

Norm on the $BV$ space:

The $BV$ space is equipped with the norm $\Vert u \Vert_{BV}=\Vert u \Vert_{L^1}+\vert u\vert_{BV}$.
Equipped with this norm the space of $BV$ functions is complete. The completeness of the $BV$ space follows from the completeness of $L^1$ and lower semicontinuity property which is stated below.

Proposition about the lower semicontinuity property of $BV-$ seminorm:
If $\{u_n\}$ is a sequence in $BV(\Omega)$ with $\sup_n |u_n|_{BV}< \infty$ that converges strongly $u_n \rightarrow u$ in $L^1(\Omega)$ then $|u|_{BV}\leq\lim \inf_{n\rightarrow \infty} |u_n|_{BV}$ and $u\in BV(\Omega)$.
Proof:
We use the following definition of BV seminorm here: $u \in L^1(\Omega)$ and $\vert u \vert_{BV}=\sup \{\int_{\Omega} u \,\mbox{ div } \mathbf{\phi}: \mathbf{\phi} \in C_c^1(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1\}<\infty$.
Let $\phi\in C_c^1(\Omega, \mathbb{R}^N)$ with $\Vert \phi \Vert_{\infty}\leq 1$.
We first observe that
$$\lim_{n\rightarrow \infty}\int_{\Omega} u_n\, \mbox{ div } \phi =\int_{\Omega} u\, \mbox{ div } \phi. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$
This follows as $u_n\rightarrow u$ in $L^1(\Omega)$ strongly, and that $\phi\in C_c^1(\Omega, \mathbb{R}^N)$. Indeed, $$ \big\vert \int_{\Omega} u_n\, \mbox{ div } \phi - u\, \mbox{ div } \phi\, \big\vert \leq \int_{\Omega} |u_n-u|\cdot |\mbox{ div } \phi | \rightarrow 0,\mbox{ as } n \rightarrow \infty.$$ Note that $ \int_{\Omega}u_n \mbox{ div } \phi \leq \sup\{\int_{\Omega} u_n \,\mbox{ div } \mathbf{\phi}: \mathbf{\phi} \in C_c^1(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1\}$, i.e. $$\int_{\Omega}u_n \mbox{ div } \phi \leq |u_n|_{BV}.$$ Taking $\lim\inf$ of both sides, we get $$\lim_{n\rightarrow \infty}\int_{\Omega}u_n \mbox{ div } \phi \leq \lim\inf_{n\rightarrow \infty} |u_n|_{BV}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$$ From (1) and (2) we get
$$\int_{\Omega}u \mbox{ div } \phi \leq \lim\inf_{n\rightarrow \infty} |u_n|_{BV}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$$
Taking supremum on the left side of (3) over all $\mathbf{\phi} \in C_c^1(\Omega, \mathbb{R}^N) \mbox{ and } \Vert \mathbf{\phi} \Vert_{\infty}\leq 1$, we get $|u|_{BV}\leq\lim \inf_{n\rightarrow \infty} |u_n|_{BV}$. Moreover, if $|u_n|_{BV}<\infty$ for all $n$ we get $|u|_{BV}<\infty$, i.e. $u\in BV$.

The weak convergence of $u_n\rightharpoonup u$ in $BV\,$ :
We say that a sequence $u_n$ in $BV(\Omega)$ converges weakly to some $u$ in $BV(\Omega)$ (i.e. $u_n\rightharpoonup u$ in $BV(\Omega)$) if and only if the following two convergences hold:
(i) $u_n\rightarrow u$ strongly in $L^1(\Omega)$
(ii) $Du_n \rightharpoonup Du$ weakly in $M(\Omega, \mathbb(R)^N)$.

Proposition: If $\{u_n\}$ is a sequence in $BV(\Omega)$ with $\sup_n |u_n|_{BV}< \infty$ that converges strongly $u_n \rightarrow u$ in $L^1(\Omega)$ then $u_n\rightharpoonup u$ in $BV$.
Proof:
As $u_n\rightarrow u$ in $L^1$ we only have to show the weak convergence $Du_n\rightharpoonup Du$.

For $\phi\in C_c^{\infty}(\Omega, \mathbb{R}^N)$ we have that $\langle Du_n, \phi\rangle =-\int_{\Omega}u_n \mbox{ div }\phi\rightarrow -\int_{\Omega}u \mbox{ div }\phi =\langle Du, \phi\rangle$.

As result of the density of $C_c^{\infty}(\Omega, \mathbb{R}^N)$ in $C_0(\Omega, \mathbb{R}^N)$ and the boundedness of $|u_n|_{BV}$ we conclude that $Du_n\rightharpoonup Du$.

APPENDIX: Basic definitions

Let $X$ be some set, and $2^X$ symbolically represent its power set, the collection of all subsets of $X$.

Measure: A mapping $\mu:2^X\rightarrow [0, \infty]$ is called a measure on $X$ if
(i) $\mu(\emptyset)=0$ and
(ii) $\mu(A)\leq \sum_{k=1}^{\infty} \mu(A_k)$, whenever $A \subset \cup_{k=1}^{\infty} A_k$.

$\mu-$ measurable set: A set $A\subset X$ is $\mu-$ measurable if for each set $B\subset X$,
$\mu(B)=\mu(B\cap A)+\mu(B\setminus A)$.


$\sigma$-algebra: A subset $\mathcal{A} \subset 2^X$ is called a $\sigma$-algebra if it satisfies the following three properties:
(i) $\mathcal{A}$ contains the null set and the set $X$, i.e. $\emptyset, X \in \mathcal{A}$.
(ii) $\mathcal{A}$ is closed under complementation: $A \in \mathcal{A}$ implies $X\setminus A \in \mathcal{A}$.
(iii) $\mathcal{A}$ is closed under countable unions: If $A_k \in \mathcal{A}$ for $k=1, 2, \dots$, then so is the union, $\cup_{k=1}^{\infty}A_k $.

Borel $\sigma$-algebra of $\mathbb{R}^n$ is the smallest $\sigma$-algebra of $\mathbb{R}^n$ containing the open subsets of $\mathbb{R}^n$.

Regular measure: A measure $\mu$ on $X$ is regular if for each set $A\subset X$ there exists a $\mu-$ measurable set $B$ such that $A\subset B$ and $\mu(A)=\mu(B)$.

Borel set: A Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

Borel measure: A measure $\mu$ on $\mathbb{R}^n$ is called Borel measure if every Borel set is $\mu-$ measurable.

Borel regular measure: A measure $\mu$ on $\mathbb{R}^n$ is called Borel regular measure if $\mu$ is Borel and for each $A \subset \mathbb{R}^n$ there exists a Borel set $B$ such that $A\subset B$ and $\mu(A)=\mu(B)$.

Radon measure: A measure $\mu$ on $\mathbb{R}^n$ is called Radon measure if $\mu$ is Borel regular and $\mu(K)< \infty$ for each compact set $K \subset \mathbb{R}^n$.