Example 21.5.36BCC82ADB7C7

Example 21.5.3.6BCC82ADB7C7 In theorem 21.5.2, vague convergence cannot be replaced by narrow convergence without a condition that prevents mass from escaping to infinity. Let $n=1$, identify $N_{\mathbb R}$ and $M_{\mathbb R}$ with $\mathbb R$ so that $N=M=\mathbb Z$, and put

Indeed, $\partial F_ j(x)$ is $\{ 0\} $ for $x<j$, $\{ 1\} $ for $x>j$, and $[0,1]$ for $x=j$. All slopes lie in $[0,1]$, and $F_ j\to 0$ locally uniformly. As in example 2.6.4, the measures $\delta _ j$ converge vaguely to $\operatorname {MA}_{\mathbb R}(0)=0$, while their total masses remain equal to one. Here each $F_ j$ has full mass for $P=[0,1]$, but the limit does not. If the functions and their limit all have full mass for one fixed body, the measures converge narrowly by lemma 21.6.2 below.