For a sequence of functionals $F_n \in C[0,1]^*$,
where
$$
F_n(f) = \int_0^1 f(x^n)dx
$$
I want to know if this sequence converges weakly.
My attempt.
I can show that there is a weak-star convergence for this sequence. This sequence weakly-star converges to $F\in C[0,1]^*$, where
$$
F(f) = f(0), \ f\in C[0,1]
$$
But I can not prove or disprove weak convergence. Using the fact that $C[0,1]^*\cong BV_0[0,1]$, where $BV_0[0,1]$ is a space of bounded variation functions $g$ such that $g(0) = 0$ and $g(x-0) = g(x)$. There is a correspondence between functionals $F_n\in C[0,1]^*$ and functions $g_n = x^{\frac{1}{n}} \in BV_0[0,1]$. And a function $g(x)$ defined by
$$
g(x) = \begin{cases}
0, & x = 0\\
1 & x\in (0, 1]
\end{cases}
$$
corresponds to $F$.
So we need to investigate the convergence $G(g_n)$ to $G(g)$ for every functional from $BV_0[0,1]^*$. I have tried many functionals but all of them converged but I can not prove this in general case. Here I stuck.
Best Answer
Consider the functional $G$ defined on $BV_0[0,1]$ by $$ G(g) = \lim_{x\to 0_+}g(x). $$ Then $ G(g_n) = 0, $ but $G(g) = 1$.