I want to prove that there exists a minimizer to the following problem $$ \min || \mu ||_{\text{TV}} \text{ such that } \mathcal{F} \mu = y $$
where $\mu \in \mathcal{M}([0,1])$, the space of Radon measures in the interval $[0,1]$, and $\mathcal{F}: \mathcal{M}([0,1]) \to \mathbb{C}^n$ is some linear operator, to which we know that $\mathcal{F}^{-1} y \neq \{ \emptyset \}$.
Typically for convex problems like this, I would approach it by using convexity and lower semi-continuity of the total variation norm. But since the underlying space is (I think) not reflexive, there is no weak compacity for closed, convex, bounded sets.
Typically, you can use that $\mathcal M(([0,1])$ is the dual space of the separable space $C([0,1])$. Hence, for every bounded sequence you can extract weak-$*$ convergent subsequences. If $\mathcal F$ is also continuous w.r.t. the weak-$*$ convergent sequences, then you can do the usual arguments. If you do not know this (additional) continuity of $\mathcal F$, the existence of minimizers may fail.