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Math Help - A question on the definability of convolution

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    A question on the definability of convolution

    In Apostol's "Mathematical Analysis", Page 328 (see the image below and the underlined sentence),

    why does the Lebesgue integral (41) exist for x\in [a,b]?
    The definition of convolution is as follows:

    Thanks!
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    Quote Originally Posted by zzzhhh View Post
    In Apostol's "Mathematical Analysis", Page 328 (see the image below and the underlined sentence),

    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)).
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