Carrying out the passage to the limit under an integral sign

Question

For a sequence of distribution functions $(F_n)$ and their characteristic functions $(\varphi_n)$ I got $$ \int_{0}^{h}{\,F_{n_k}(y)\,dy}\,-\,\int_{-h}^{0}{\,F_{n_k}(y)\,dy}\,\,=\,\,\frac{1}{\pi}\,\int_{-\infty}^{\infty}{\,\frac{1\,-\,\cos{ht}}{t^2}\,\varphi_{n_k}(t)\,dt}. $$ It should be easily seen that the passage to the limit, $k\to\infty$, can be carried out under the integral signs so that $$ \frac{1}{h}\,\int_{0}^{h}{\,F(y)\,dy}\,-\,\frac{1}{h}\,\int_{-h}^{0}{\,F(y)\,dy}\,\,=\,\,\frac{1}{\pi}\,\int_{-\infty}^{\infty}{\,\frac{1\,-\,\cos{y}}{y^2}\,\varphi\left(\frac{y}{h}\right)\,dy}. $$ But, why?

Answer 1

$$\left|\frac{1-\cos ht}{t^2}\right|\underbrace{|\varphi_{n_k}(t)|}_{\leq 1}\leq \left|\frac{1-\cos ht}{t^2}\right|\in L^1(-\infty ,\infty )$$ therefore dominated convergence theorem can conclude.