Total variation inequality liminf

Question

I'm trying to understand the proof, given by t.b. here Space of Complex Measures is Banach (proof?) .

In the course of the proof, the author uses the equality

$\left|\left| \mu - \mu_{m} \right|\right| \leq \liminf\limits_{n \to \infty} \left|\left|\mu_{n}-\mu_{m}\right|\right|$,

where $\left|\left| \cdot \right|\right|$ denotes the total variation norm and

$\mu(A):=\lim\limits_{n \to \infty} \mu_{n}\left(A\right)$

the limit of a sequence of measures $\mu_{n}$ with bounded total variation.

I would like to know, why the above inequality holds. I tried several thinks using the definition of the total variation and I came up with nothing.

Any help is most appreciated.

Is somewhere proven that the variational norm is lower semi continuous w.r.t. to the convergence you mentioned? — Quickbeam2k1, Sep 26 '14 at 15:01

score 3 · Accepted Answer · answered Sep 26 '14 at 15:02

It suffices to show that $\|\mu\| \le \liminf_n \|\mu_n\|$. The general case follows quickly from that.

Take any finite disjoint collection $\{E_k\}_{k=1}^m$ of measurable sets. Then $$\sum_{k=1}^m \mu(E_k) = \lim_{n \to \infty} \sum_{k=1}^m |\mu_n(E_k)|.$$ But $$\sum_{k=1}^m |\mu_n(E_k)| \le \|\mu_n\|$$ by the definition of variation. Take the limit inferior on both sides to get $$\lim_{n \to \infty} \sum_{k=1}^m |\mu_n(E_k)| \le \liminf_{n \to \infty} \|\mu_n\|.$$ Now take the supremum over all such collections $\{E_k\}$ to obtain $\|\mu\| \le \liminf \|\mu_n\|$.

Thanks a lot for this insight. – Blasco Sep 26 '14 at 15:13 — Blasco, Sep 26 '14 at 15:13

score 0 · Answer 2 · answered Dec 10 '20 at 09:07

We will use the notation from the proof that was mentioned.

By defintion, $| \mu_n(E) - \mu_m(E) | \le || \mu_n - \mu_m ||$, hence $\liminf_n | \mu_n(E) - \mu_m(E) | \le \liminf_n || \mu_n - \mu_m ||$.

It follows that $| \mu(E) - \mu_m(E) | \le \liminf_n || \mu_n - \mu_m ||$.

Taking supremum over E, we have $|| \mu - \mu_m || \le \liminf_n || \mu_n - \mu_m ||$, q.e.d.

Total variation inequality liminf

2 Answers2