Help with qual problem on Riemann-Stieltjes Integration

  1. Let f:[0, ∞) → ℝ be a decreasing function. Assume that

    |[itex]\int^{\infty}_{0}[/itex]f(x)dx| < ∞

    Show that

    [itex]lim_{x→∞}[/itex]xf(x) = 0.


    Attempt:
    By hypothesis f is integrable for every b>0, as is the function x, so that for every such b, xf(x) is integrable on [0,b]. From here I tried passing to sums by considering a partition P such that U(P,xf(x)) - L(P,xf(x)) < ε, but I'm probably barking up the wrong tree there. I also can't see a way to use integration by parts to generate an xf(x) term since I don't have differentiability or continuity of f.

    I'm pretty much just lost and need a slight nudge in the right direction.
     
  2. jcsd
  3. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    No, integration by parts can't help, for just the reason you mention. lim f(x) as x->infinity has to be zero, right? That should be easy to see. So you can assume lim f(x)>0. Prove it by contradiction. So suppose lim x->infinity x*f(x) is NOT zero. That means there is an e>0 such that for any N>0 there is c such that c*f(c)>e. That should be enough of a slight nudge to get you started.
     
  4. Thanks! that was just what I needed.
     
  5. I thought I was good, but I'm having trouble seeing where the contradiction lies. I expected that maybe I could choose N large so that f(x)<ε, yet x*f(x)>ε, but I can't see how will work. I know that ∫x*f(x) need not converge so there's nothing there. I also looked at the same N large under the same conditions to see if choosing a partition P such that U(P,x*f(x))-L(P,x*f(x))<ε might imply x*f(x)<ε, but that didn't work either. I must be making dunce move and missing something super obvious here. :(
     
  6. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    Ok, pick a c such that c*f(c)>e. That means the integral of f(x) from 0 to c is greater than e, right? Now pick a c'>2c such that c'*f(c')>e. Can you get a lower bound for the integral from c to c' of f(x)?
     
  7. Ah! Got it! I was completely missing out on the fact that f decreasing implies x*f(x) < ∫f dx. Kept focusing too hard on other irrelevant things. Thanks!
     
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