Notice so by Tonelli's theorem then a standard theorem tells you that the convolution is almost everywhere finite, but since (it's easy to prove) it is continous it cannot blow up at any number (only at plus/minus infinity).
Show that if is locally integrable and that , then .
OK, so first thing I'm thinking to do is to use the commutativity of the convolution operator and then say since has compact support (which in this setting means it vanishes at positive and negative infinity, or that it has support on a bounded interval), we can write the convolution in terms of a finite integral in this way:
For some interval .
And now, trying to figure out the bounds has me confused, since we know nothing about other than local integrability (so we can't say that we have less than or equal to or anything of that sort), so I'm having issues. :-/