When can we write $\lim_{n\to\infty}\sum_{j=1}^\infty e^{ij}f(\frac jn)\frac1n=\lim_{n\to\infty}\int_0^\infty e^{inx}f(x)dx$, for integrable $f$?

Let $f_2$ be an integrable function. I am trying to sum

$$\lim_{n\rightarrow\infty}\sum_{j=1}^{\infty}e^{ij}f_{2}\left(\frac{j}{n}\right)\frac{1}{n}$$

Under what condition can I write this as

$$\lim_{n\rightarrow\infty}\int_{0}^{\infty}e^{inx}f_2(x)dx$$

Is this justified in any sense?