In this book (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here:

Let $f:I\times E\rightarrow E$ a Pettis integrable function, where $I:=[0,T]\subset \mathbb{R}$, and $E$ is a Banach space. Let $\Omega$ be a bounded, equicontinuous subset of $\mathcal{C}(I,E)$.

Suppose that $f(.,y(.)),\;y\in \Omega$ is equicontinuous.

Then, the integrals of these functions $\int_{0}^{t}f(s,y(s))ds,\;y\in \Omega$ can be

uniformly approximatedby integral sums $$\frac{t}{n} \sum_{i=1}^{n} f\left(s_{i}, y\left(s_{i}\right)\right), \quad s_{i}=i \frac{t}{n}, y \in \Omega $$

My first question is: what this "uniformly approximated" refers to?

Secondly, I am looking for a proof of this result, and it will be great if someone give me a reference to include in an article.

**EDIT:** As pointed by @Jochen Wengenroth in comments, this is not true in general. In the book they suppose that $f(.,y(.)),\;y\in \Omega$ is also equicontinuous, I forgot to mention that, and i'm sorry for that!