I am trying to understand the proof sketch of the following.
Let $\kappa$ and $(\kappa_n)_{n\geq 1}$ be a measure and a sequence of finite measures defined on the borelians of $\mathbb{R}^p$ such that \begin{equation}\label{unifb}\tag{1} \kappa(\mathbb{R}^p)< \infty, \quad\kappa_n(\mathbb{R}^p)\leq C ,\quad \forall n \end{equation} Denote the set of continuity sets of $\kappa$ as $\mathcal{C}_\kappa$ - i.e. $E\in \mathcal{C}_\kappa$ if $\kappa(\partial(E))=0$, where $\partial(E)$ is the Boundary of $E$.
Besides (\ref{unifb}), I have other properties:
- $\lim_{n \to \infty}\kappa_n(E) = \kappa(E), \quad (E \in \mathcal{C}_\kappa, 0 \notin \bar{E} \hbox{ and } E\, \hbox{ bounded } )$
- $\int_{|x|\geq\epsilon}d\kappa_n \to \int_{|x|\geq\epsilon}d\kappa, \quad \forall \, \epsilon>0$ such that $\{x : |x| \geq \epsilon\} \in \mathcal{C}_{\kappa} $
I want to show that \begin{equation}\label{unifb2}\tag{2} \lim_{n \to \infty}\kappa_n(E) = \kappa(E), \quad (E \in \mathcal{C}_\kappa, 0 \notin \bar{E} ) \end{equation}
Proof Sketch
If $E$ is bounded, no problem for me. So I will assume $E$ unbounded.
I will list two facts that the author argues to demonstrate (\ref{unifb2}) and that I would like to understand better.
- I) Since $\kappa(\mathbb{R}^p) < \infty$, $\int_{|x|\geq\epsilon}d\kappa_n$ and $\int_{|x|\geq\epsilon}d\kappa$ converges uniformly to $0$ as $\epsilon \to \infty$.
- II) The property 1 above and the fact I implies (\ref{unifb2})
I understand well that $\lim_{\epsilon \to \infty} \int_{|x|\geq\epsilon}d\kappa_n = 0$ and $\lim_{\epsilon \to \infty} \int_{|x|\geq\epsilon}d\kappa = 0$. But how does this happen uniformly as a consequence of $\kappa(\mathbb{R}^p) < \infty$ ?
My attempt to prove (\ref{unifb2}) is: given $E \in \mathcal{C}_\kappa, 0 \notin \bar{E} $
$$\kappa_n(E) = \kappa_n(\underbrace{E \, \cap [|x| < \epsilon]}_{:=E_1^\epsilon} ) + \kappa_n(\underbrace{E \, \cap [|x| \geq \epsilon]}_{:=E_2^\epsilon} )$$
Note that $E_1^\epsilon$ is bounded, $E_1^\epsilon \in \mathcal{C}_\kappa$ and $0\notin \bar{E_1^\epsilon}$ . Thus using property 1 above, we have $\kappa_n(E_1^\epsilon) \to \kappa(E_1^\epsilon)$.
Now, I want to use property 2 and the fact I in order to work with $\kappa_n(E_2^\epsilon)$ and show that $\kappa_n(E_2^\epsilon) \to \kappa(E_2^\epsilon)$.
With these two results I think I can show (\ref{unifb2}).
Some help please