2
$\begingroup$

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, t]) = v(t),~ t\in [0, 1]$. Since the set of signed measures can be identified with the dual of the Banach space $\mathcal{C}([0, 1], R)$, $V$ can be equipped with the weak-$\ast$ topology. Convergence of elements of V in this topology will be denoted by $\stackrel{w}{\to}$ . In particular, $v^n\stackrel{w}{\to}v^0$ means that for every continuous function $f\in \mathcal{C}([0, 1], R)$ one has

$$\lim_{n\to\infty}\int_{[0,1]}f(t)dv^n(t)=\int_{[0,1]}f(t)dv^0(t)$$

Now given a function $\xi: V\to R$ and a set $\Lambda\subset \mathcal{C}([0, 1], R)$ of increasing piecewise affine functions $\lambda$ with $\lambda(0)=0$ and $\lambda(1)=1$. Assume that for any $\varepsilon>0$ one may find $\delta>0$ such that for any $\lambda\in\Lambda$ satisfying $||\lambda-I||\le \delta$ one has

$$|\xi(v)-\xi(v\circ\lambda)|\le \varepsilon,~ \forall v\in V.$$

Here $I$ dentoes the identity function and $\circ$ denotes the composition of functions. My question is could we show (or maybe under some supplementary conditions) that $\xi$ is continuous with respect to $\stackrel{w}{\to}$? Thx for the reply

Improve this question
$\endgroup$
7
  • $\begingroup$ I am looking for some reference characterizing the continuity/upper semicontinuity of functions with respect to weak-$\ast$ topology. $\endgroup$
    – CodeGolf
    Commented Nov 13, 2014 at 10:14
  • $\begingroup$ Do you want $\Lambda$ to be a set of increasing piecewise affine functions? The only affine function $\lambda \colon [0,1] \to \mathbb{R}$ such that $\lambda(0)=1$ and $\lambda(1)=1$ is the identity. I think more constraints on $\Lambda$ are needed: for example at the moment I think that $\Lambda$ could be empty yet still meet the above set of requirements. $\endgroup$
    – Ian Morris
    Commented Nov 13, 2014 at 10:21
  • $\begingroup$ Yes, $\lambda$ should be \textit{piecewise} affine, thx for this comment $\endgroup$
    – CodeGolf
    Commented Nov 13, 2014 at 10:23
  • $\begingroup$ In fact, the condition above is a condition that i imagine that $\xi$ should satisfy, but i am not sure at all it is the good condition to ensure the continuity of $\xi$. I am seeking the suitable conditions. $\endgroup$
    – CodeGolf
    Commented Nov 13, 2014 at 10:26
  • $\begingroup$ As far as I know, i can find some usual functions which are continuous or semicontinuous w.r.t. this topology. For example, $v\to \int_0^1v(t)\eta(t)$, where $\eta$ is an atomeless measure on $[0,1]$. Moreover, we know that $v^n\stackrel{w}{\to}v^0$ implies that there exists a countable set $T\subset [0,1)$ such that for every $t\in [0,1]\backslash T$ one has $v^n(t)\to v^0(t)$. Thus it easy to show $\liminf_{n\to\infty}||v^n||\ge ||v^0||$, which implies that $v\to ||v||$ is lower semicontinuous. $\endgroup$
    – CodeGolf
    Commented Nov 13, 2014 at 10:36

0

Reset to default

You must log in to answer this question.