Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f,g\in L_1(\mathbb R, m)$ where m is the Lebesgue measure.

Prove that:

a) $f(x-t)g(t) \in L_1(\mathbb R, m)$ as a fuction of $t$ almost all $x$

b) $h\in L_1(\mathbb R, m)$ where $h=(f \ast g)(t)=\int_{\mathbb R}f(x-t)g(t)dm(t)$

c)$||h||_{L_1}\le ||f||_{L_1}||g||_{L_1}$

share|cite|improve this question

1 Answer 1

up vote 3 down vote accepted

Hint: Use Fubini on $\mathbb{R}^2$ for the function $G(x,t)=|f(x-t)g(t)|$.

If you need more help, see p.16 here:

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.