The following is extracted from 'Fourier Analysis on Groups' by Rudin, page $4,$ under the section 'Convoluton':
LCA denotes locally compact abelian.
For any pair of Borel functions $f$ and $g$ on the LCA group $G,$ define their convolution $f * g$ by the formula $$(f*g)(x) = \int_G f(x-y)g(y) dy,$$ provided that $$\int_G |f(x-y) g(y)|dy < \infty.$$
Question: Why do we need $$\int_G |f(x-y) g(y)|dy < \infty$$ in the definition of convolution? W
What will happen if we defined convolution without it?