Example 2.6.481B4A3D092FF

Example 2.6.4.81B4A3D092FF On $\mathbb R$, the Dirac measures $\delta _ j$, $j\geq 1$, converge vaguely to zero, since every $f\in C_ c(\mathbb R)$ vanishes at $j$ for large $j$; yet each has total mass one. On $\Omega =(0,1)$ the measures $\delta _{1/j}$, $j\geq 2$, behave in the same way, the mass escaping through the boundary point $0$, which does not belong to $\Omega $. Neither sequence is eventually uniformly tight: a compact subset of $\mathbb R$ does not contain $j$ for large $j$, and a compact subset of $(0,1)$ does not contain $1/j$ for large $j$, so the mass outside a fixed compact set is eventually equal to one.