integration limit in convolution

Question

Assume that I have two functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$. I want to consider its convolution, but only for $x$ from a set $A\subset\mathbb{R}$. What is true

$(f*g)(x)=\int_{\mathbb{R}}f(x-y)g(y)\,dy$

or

$(f*g)(x)=\int_{A}f(x-y)g(y)\,dy$?

Answer 1

The first one. Convolution is defined in terms of an integral of $y$ over the entire axis, regardless of the number at which you evaluate it.