I found in a textbook this definition of expectation for a random variable $X$
$$E(X)=\lim_{n\to\infty}\sum_{i=1}^n\frac{i-1}{\sqrt{n}}P\left(\frac{i-1}{\sqrt{n}}<X\leq \frac{i}{\sqrt{n}}\right)$$
However, the textbook doesn't particularly mention where this definition comes from, when it's from, and how it's equivalent to the current definition to expectation. Also, I cannot find any other references to this anywhere. If anyone knows anywhere to further read about this, it would be highly appreciated. Thanks!