Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 64 / 3

The structure theorem for functions of bounded variation is the measure representation of the distributional derivative: for $u\in BV(\Omega )$ there is a unique finite vector Radon measure $Du$ withfor every $\phi \in C_c^1(\Omega ;{\mathbb{R}}^n)$. It also has a polar decomposition of a vector measure $Du={\sigma }_u|Du|$ with $|{\sigma }_u|=1$ for $|Du|$-almost every point.

To prove it, let $L(\phi )=-\int u\mathrm{div}\phi$. The test-function definition of total variation seminorm, applied to both signs, gives $|L(\phi )|\le \mathrm{TV}(u)\Vert \phi {\Vert }_{\infty }$. Compactly supported smooth vector fields are uniformly dense in $C_0(\Omega ;{\mathbb{R}}^n)$, so $L$ extends uniquely to a bounded functional there. The Riesz-Markov-Kakutani representation theorem, applied componentwise, gives the unique vector measure $Du$. The operator norm of $L$ is exactly the defining variation supremum; the vector-measure dual norm is $|Du|(\Omega )$, proving equality. Conversely any finite measure satisfying the identity bounds that supremum, so this also characterizes membership in the BV space.

The Radon-Nikodym theorem applied to the components relative to $|Du|$ gives ${\sigma }_u$; the definition of the variation measure forces $|{\sigma }_u|=1$ almost everywhere. If $u$ is smooth, ordinary integration by parts gives $Du=\nabla u{\mathcal{L}}^n$. Compact support of the test field removes any boundary contribution; no boundary regularity is needed for this representation statement.

First apply Lebesgue decomposition to the derivative measure relative to ${\mathcal{L}}^n$:Here $\nabla u$ is the almost-everywhere approximate gradient, rather than an assertion that $u$ belongs to $W^{1,1}$. Split the singular part into its jump part of a BV derivative and Cantor part of a BV derivative:An approximate jump point has a unit normal ${\nu }_u$ and distinct finite BV traces on a hypersurface $u^{+},u^{-}$, obtained as mean limits on the corresponding two half-balls. Their set $J_u$ is the jump set of a BV function, countably $(n-1)$-rectifiable. Reversing the normal swaps the BV traces on a hypersurface and leaves the displayed measure unchanged. The approximate discontinuity set $S_u$ differs from $J_u$ only by an ${\mathcal{H}}^{n-1}$-null set.

The remaining $D^{c}u=D^{s}u-D^{j}u$ is singular to Lebesgue measure and gives zero mass to every set with sigma-finite ${\mathcal{H}}^{n-1}$ measure. It is diffuse rather than a second jump contribution. In one dimension the three parts are illustrated by an affine function, a step function and the Cantor function, respectively. Countably many jumps therefore do not imply that the singular derivative has no Cantor part of a BV derivative.

Put $\mu =Du$, a finite signed Radon measure, and defineFor $\phi \in C_c^1((a,b))$, Fubini's theorem for the signed measure givesThus $D{F}_l=\mu =Du$. A distribution on a connected interval with zero derivative is constant: any compactly supported test of integral zero is the derivative of a compactly supported test, so it pairs to zero with $u-F_l$. Choose the resulting constant $c$. Thenare one-sided representatives of a one-dimensional BV function. The first equals $u$ almost everywhere. Their difference is $\mu (\{t\})$, nonzero at at most countably many points, so the second also equals $u$ almost everywhere.

Finite-measure continuity applied to $|\mu |$ proves $F_l(s)\to F_l(t)$ as $s\uparrow t$, and $F_r(s)\to F_r(t)$ as $s\downarrow t$. Moreover the opposite one-sided limits are $F_l(t+)=F_r(t)$ and $F_r(t-)=F_l(t)$. Consequently both representatives are continuous exactly where $\mu (\{t\})=0$. For each positive integer $m$, there are only finitely many measure atoms of magnitude at least $1/m$, since their total magnitudes are bounded by $|\mu |((a,b))$. Their union is countable, proving the requested discontinuity bound. This does not require the jump points to be isolated; they can be dense.

The hint's one-dimensional statement is consistent with this construction: zero ${\mathcal{H}}^0$ measure means an empty set, so $S_u\subseteq J_u$. This concerns approximate discontinuities of the BV space class; arbitrary changes of representative at points could create artificial pointwise discontinuities. The interval-mass representatives above remove that ambiguity and give the required one-sided continuity directly.