If an application $f$ has a known density (for example, $f\sim\mathcal{N}(0,1)$), does the following equality stand : $\int_{-\infty}^{+\infty}f(x) \; \textrm{d}x = \int_{-\infty}^{+\infty}f(x) \; \textrm{pdf}(f(x)) \; \textrm{d}x$ ?
If yes: how to prove this ? If not: is it possible to quantify the error which is introduced ?
