A question on the definability of convolution

• Feb 17th 2010, 02:18 PM
zzzhhh
A question on the definability of convolution
In Apostol's "Mathematical Analysis", Page 328 (see the image below and the underlined sentence),
http://i47.tinypic.com/f0n8gi.jpg
why does the Lebesgue integral (41) exist for $x\in [a,b]$?
The definition of convolution is as follows:
http://i50.tinypic.com/2b4rqr.jpg
Thanks!
• Feb 19th 2010, 04:58 AM
Laurent
Quote:

Originally Posted by zzzhhh
In Apostol's "Mathematical Analysis", Page 328 (see the image below and the underlined sentence),
http://i47.tinypic.com/f0n8gi.jpg
why does the Lebesgue integral (41) exist for $x\in [a,b]$?

One should probably replace [a,b] by [0,a].
Then this is just because the product of two Riemann-integrable functions is Riemann-integrable (note that a Riemann-integrable function on [0,a] is bounded, contrary to a Lebesgue-integrable function on [0,a]). And if g is Riemann-integrable on [0,a], then so is $t\mapsto g(x-t)$ for any $x\in[0,a]$, because it is obtained by symmetry from $g$, which is R-integrable on $[x-a,x]\subset (-\infty,a]$ (remember g is zero on $(-\infty,0)$).