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 $\{f_n\}$?