Is the following equation established?
It's a version of Parseval's Theorem, which can be easily proved by noting that

$$\int_{-\infty}^{\infty}s(t)r^*(t)dt=\mathcal{F}\{s(t)r^*(t)\}\big|_{f=0}\tag{1}$$

where $\mathcal{F}$ denotes the Fourier transform. Note that in general the integration limits should be $-\infty$ and $\infty$ for both integrals. I assume your notation refers to band-limited functions.

From $(1)$, the proof is quite short:

$$\mathcal{F}\{s(t)r^*(t)\}=S(f)\star R^*(-f)=\int_{-\infty}^{\infty}S(\nu)R^*(\nu -f)d\nu\tag{2}$$

where $\star$ denotes convolution. So

$$\mathcal{F}\{s(t)r^*(t)\}\big|_{f=0}=\int_{-\infty}^{\infty}S(\nu)R^*(\nu)d\nu\tag{3}$$

which establishes the equation in your question.