Consider a function $f \in \mathcal L^1 [0, 1]$: we define the total variation of $f$ as usual by $$ V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |, $$ where the supremum is taken over the set of partitions $\{ x_0, \ldots, x_n \}$, with $n \ge 1$ and $0 \le x_0 < \ldots < x_n \le 1$.
Now, for an equivalence class $f \in L^1$, we define $V_0^1 (f) = \inf_{g \in f} V_0^1 (g)$. I was wondering if, likewise to the multidimensional case, one could establish lower semicontinuity of the total variation in $L^1$, that is, $$V_0^1 (f) \le \liminf_n V_0^1 (f_n)$$ where $f_n \in L^1 [0, 1]$, $f_n \to f$ in $L^1$ and there exists some $M > 0$ s.t. $V_0^1 (f_n) \le M$ for all $n$.
My first intuition was to use an alternate expression of total variation that would more closely ressemble the definition used for multivariate functions. If $f$ is an integrable function and $V_0^1 (f) < \infty$, we define a continuous linear functional on $\mathcal C^0 [0, 1]$ by $\phi \mapsto \int_0^1 \phi df$, where the integral is a Riemann-Stieltjes integral w.r.t. $f$. Now, since that expression may not be well-defined in the case of an $f \in \mathcal L^1$ of unbounded variation, we shall restrict ourselves to $\phi \in \mathcal C^1 [0, 1]$, and the total variation of $f$ can then be written as $$V_0^1 (f) = \sup \left\{ \int_0^1 \phi (t) d f (t) : \phi \in \mathcal C^1 [0, 1], {\| \phi \|}_\infty \le 1 \right\}$$ Integration by parts allows us to further rewrite this integral as $$ \int_0^1 \phi (t) d f (t)=\phi(1) f(1) - \phi(0) f(0) - \int_0^1 f \phi'. $$ Now, in the case of an integrable equivalence class $f$, I am hoping we can obtain an expression in the likes of the following conjecture: $$V_0^1 (f) \overset{?}{=} \sup \left\{ \int_0^1 f (t) \phi' (t) dt : \phi \in \mathcal C^1 [0, 1], {\| \phi \|}_\infty \le 1 \right\},$$ which would be similar to the multivariate case. The problem lies in the boundary terms $\phi(0) f(0)$ and $\phi(1) f(1)$ that appear with integration by parts, as they depend on the version of $f$ chosen and may prove tricky to get rid of once we take the infimum over all versions of $f$.
Is it possible to prove (or disprove) the above conjecture?
Consider a subsequence $\{f_{n_k}\}_k$ such that $$\liminf_nV_0^1(f_n)=\lim_kV_0^1(f_{n_k})$$ Find representatives $g_k$ of $f_{n_k}$ such that $V^1_0(g_k)\le V^1_0(f_{n_k})+1/k$. By taking a further subsequence, we can assume that $g_k$ converges a.e. Since $g_k$ can be written as the difference of two increasing functions both uniformly bounded in $k$, by applying the Helly selection principle, we can find a further subsequence, not relabeled, such that $g_k$ converges pointwise to some function $g$, which will be a representative of $f$. Consider a partition . By pointwise convergence $$\sum_i|g(x_i)-g(x_{i-1})|=\lim_k\sum_i|g_k(x_i)-g_k(x_{i-1})|\le\liminf_kV^1_0(g_k)$$ Take the supremum over all partitions.