As I understand it, to convolve $f$ and $g$ means to find $\displaystyle \int_{\mathbb R} f(a)g(t-a)da$, which is also apparently commutative, and therefore $\displaystyle \int_{\mathbb R}f(a)g(t-a)da = \displaystyle \int_{\mathbb R}f(t-a)g(a)da$
That means, if $f(a) = 1$, then $\displaystyle \int_{\mathbb R}g(t-a)da = \displaystyle \int_{\mathbb R}g(a)da$. For $g(t) = t$, for example, is not true. So what am I misunderstanding?