Signed narrow convergence controls variation tails and Bochner moments

The supremum-norm space $C_b(X)$ is Banach: a uniformly Cauchy sequence converges pointwise, hence uniformly, and a uniform limit of continuous functions is continuous. A signed measure $\sigma$ defines a bounded functional $T_{\sigma }f=\int fd\sigma$ of norm $\mathrm{\mid }\sigma \mathrm{\mid }(X)$. The upper bound is the variation inequality. For the reverse bound, use [F1] and [F2] to choose disjoint compact subsets of the two Hahn pieces with omitted variation arbitrarily small. The distance formula separating these compacts gives a bounded continuous function equal to $1$ and $-1$ there, with magnitude at most one, whose integral approaches $\mathrm{\mid }\sigma \mathrm{\mid }(X)$. Since $T_{{\sigma }_n}f$ converges for every $f$, [F3] gives ${\sup }_n\mathrm{\mid }{\sigma }_n\mathrm{\mid }(X)<\mathrm{\infty }$.

Put $y_n={\sigma }_n-\nu$ and suppose their variations are not uniformly tight. Some $ϵ>0$ then remains outside every compact ball for an arbitrarily late index: each fixed finite set of indices is tight by [F2]. Recursively choose $n_j$ increasing, radii $R_j\to \mathrm{\infty }$, and $q_j\in C_c(X)$ with disjoint supports in $\{R_{j-1}<\mathrm{\mid }x\mathrm{\mid }<R_j\}$, $\mathrm{\mid }{q}_j\mathrm{\mid }\le 1$, and $\int q_{j}d{y}_{n_j}\ge ϵ\mathrm{/}2$. At stage $j$, choose an index threshold so late that $\mathrm{\mid }\int {\sum }_{i<j}{q}_{i}d{y}_n\mathrm{\mid }<ϵ\mathrm{/}8$ thereafter, using narrow convergence. Enlarge the exclusion ball to make every earlier-index variation tail smaller than $ϵ$, and select a later index with tail at least $ϵ$. Regularity of its positive and negative Hahn pieces and distance cutoffs with compact support in that exterior produce $q_j$ with the required integral. Finally choose $R_j$ larger than its support and with $\mathrm{\mid }{y}_{n_j}\mathrm{\mid }(X\setminus B_{R_j})<ϵ\mathrm{/}8$. Increasing radii by one leaves gaps between the annuli. The locally finite sum $q={\sum }_j{q}_j$ lies in $C_b(X)$ with $\mathrm{\mid }q\mathrm{\mid }\le 1$. Its pairing with $y_{n_j}$ is at least $ϵ\mathrm{/}2-ϵ\mathrm{/}8-ϵ\mathrm{/}8=ϵ\mathrm{/}4$: the earlier terms are small by construction and the entire later sum is bounded by the chosen tail. This contradicts narrow convergence to zero. Thus $(\mathrm{\mid }{y}_n\mathrm{\mid })$ is uniformly tight. Since $\mathrm{\mid }{\sigma }_n\mathrm{\mid }\le \mathrm{\mid }{y}_n\mathrm{\mid }+\mathrm{\mid }\nu \mathrm{\mid }$ and the fixed finite measure $\mathrm{\mid }\nu \mathrm{\mid }$ is tight by [F2], $(\mathrm{\mid }{\sigma }_n\mathrm{\mid })$ is uniformly tight as well.

Given $\delta >0$, choose compact $K$ with ${\sup }_n\mathrm{\mid }{\sigma }_n\mathrm{\mid }(X\setminus K)+\mathrm{\mid }\nu \mathrm{\mid }(X\setminus K)<\delta$. On $K$ a finite sum ${\varphi }_{ϵ}={\sum }_{i=1}^N{h}_i{\psi }_i$, with ${\psi }_i\in C_c(X)$, approximates $\varphi$ uniformly within $ϵ$, and can be chosen with $\mathrm{\parallel }{\varphi }_{ϵ}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}$. To construct it, cover $K$ by finitely many relative balls on which $\varphi$ varies by less than $ϵ$, use distance cutoffs, normalize their sum near $K$, and multiply by a compact cutoff equal to one on $K$; choose $h_i$ as feature values at the centers. Its integral converges in norm by scalar narrow convergence of the finitely many ${\psi }_i$. The remaining norm error is at most $ϵ(\mathrm{\mid }{\sigma }_n\mathrm{\mid }(X)+\mathrm{\mid }\nu \mathrm{\mid }(X))+2\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}(\mathrm{\mid }{\sigma }_n\mathrm{\mid }(X\setminus K)+\mathrm{\mid }\nu \mathrm{\mid }(X\setminus K))$. Let $ϵ,\delta \downarrow 0$ to prove the Bochner-moment conclusion; [F4] gives existence of all integrals.

The restrictions to $K$ of the bounded-Lipschitz unit ball have finite uniform $ϵ$-nets. Explicitly take a finite $ϵ$-net of points of $K$, round each function's values there to a rational grid in $[-1,1]$, and use its Lipschitz bound to control the error between those points. Only finitely many rounded value patterns occur; retain one actual function for each occurring pattern. Narrow convergence controls their finitely many pairings, while step 1.1 bounds the uniform approximation error on $K$ and step 2.1 bounds all tail errors. Letting the mesh and tail bound decrease proves BL convergence.

For $0\le q\le \psi$ with $q\in C_c(X)$, $\int \psi d{\sigma }_n^{+}\ge \int qd{\sigma }_n\to \int qd\nu$. The supremum of the last quantity over such $q$ equals $\int \psi d{\nu }^{+}$: use the Hahn pieces, compact regular approximation, and distance cutoffs to approximate their sign restriction. Similarly, for $q\in C_c(X)$ with $\mathrm{\mid }q\mathrm{\mid }\le W$, $\int Wd\mathrm{\mid }{\sigma }_n\mathrm{\mid }\ge \int qd{\sigma }_n\to \int qd\nu$, and the supremum is $\int Wd\mathrm{\mid }\nu \mathrm{\mid }$. For the lower bound in this supremum, multiply $W$ by a compact cutoff and a continuous function of magnitude at most one that is $1$ and $-1$ on compact subsets of the two Hahn pieces; omitted weighted variation tends to zero by [F2]. These prove the two lower-semicontinuity claims without assuming convergence of Jordan parts. Any failure of a continuous-time moment or BL limit yields a sequence $t_n\to \mathrm{\infty }$ contradicting the result just proved.