Question

A collection ${f_n}$ of real valued functions is said to be HK-equi-integrable on $I=[a,b]$, if there exists a gauge $\delta$ on $I$ such that for every $\epsilon>0$, there exists a $\delta$-fine tagged partition $\mathcal{P}$ of $I$ such that $\|S(f,\mathcal{P})-\int_I f\|_X<\epsilon$ for every $n$. That is, the gauge $\delta$ is independent on $n$.
Where gauge $\delta$ is any positive function $\delta:[a,b]\rightarrow\mathbb{R}^+$. And $S(f,\mathcal{P})=\sum_i f(\xi_i)(x_i-x_{i-1})$.

Now my question is that does equi-integrability of ${f_n}$ and point wise convergence of ${f_n}$ to $f$ implies uniform convergence of the sequence ${f_n}$?

Answer 1

What if you try these functions $f_n$ on $[0,1]$ ... piecewise linear, with $f_n(0) = 0$, $f_n(1/n)=-1$, $f_n(2/n)=1$, $f_n(3/n)=0$, $f_n(1)=0$.