Convolution for two random variables

Question

In the textbook i'm currently reading it is said that for two independent random variables $X$ and $Y$ density function of variable $Z=X+Y$ can be found from the equation: $$ g(z) = \int_{-\infty}^\infty f_1(x)f_2(z-x) \, dx $$ And if possible values of arguments are positive, then $$ g(z) = \int_0^z f_1(x)f_2(z-x) \, dx $$ Why the upper bound $\infty$ has been replaced with argument $z$ at the second ingeral?

Answer 1

If the possible values (range of the two r.v.'s) are non-negative, then both $x$ and $z-x$ should be non-negative in the integrand (as $f_1(x) = 0$ and $f_2(z-x)=0$ otherwise, respectively). This implies $x\geq 0$ and $x\leq z$.

Hence the bounds $\int_0^z$ on the integral.