Mass normalization in mollified singular integrals ($T({\varphi }_{\epsilon })\ast f={\varphi }_{\epsilon }\ast T(f)⟶(\int \varphi )T(f)$)

For a bounded Fourier multiplier operator $T$ on $L^2$ and a bump function $\varphi$, the displayed convolution identity follows by multiplying Fourier transforms. The convergence holds in $L^2$ and almost everywhere by the approximate identity theorem, after accounting for the integral of the kernel. The limit is $T(f)$ when $\int \varphi =1$. Smoothness and compact support alone do not imply this normalization.