How to show $ |f*g|_{1} \le |f|_{1}|g|_{1}$

Question

I have to show $ \|f*g\|_1 \le \|f\|_1\|g\|_1$.

Answer 1

I will assume that $f,g: \Bbb R^d \to \Bbb R$. Observe:

$$\int_{\Bbb R^d} \left| \int_{\Bbb R^d} f(x-y)g(y) dy \right|dx \le \int_{\Bbb R^d} \left[ \int_{\Bbb R^d} |f(x-y)||g(y)|dy \right] dx$$

Using Tonelli's theorem, the RHS is equal to

$$\int_{\Bbb R^d} \left[ \int_{\Bbb R^d} |f(x-y)||g(y)|dx \right] dy$$

Which after rearrangement becomes $\|f\|_1 \|g\|_1$. Hence $f * g \in L^1(\Bbb R^d)$ and $\|f * g\|_1 \le \|f\|_1 \|g\|_1$