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Math Help - Is sin(1/x) measurable?

  1. #1
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    Is sin(1/x) measurable?

    I'm asked to find the Lebesgue integral of sin (1/x) on the interval (0,1], if it exists. But is sin(1/x) even a measurable function? It seems to me that the answer is no, but I cannot prove this.
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  2. #2
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    Quote Originally Posted by Amanda1990 View Post
    I'm asked to find the Lebesgue integral of sin (1/x) on the interval (0,1], if it exists. But is sin(1/x) even a measurable function? It seems to me that the answer is no, but I cannot prove this.
    It is continuous on (0,1], hence it is of course measurable. In addition, it is bounded, so that it is Lebesgue-integrable on this bounded interval (Indeed, \left|\sin\frac{1}{x}\right|\leq 1 and x\mapsto 1 is integrable on (0,1], so that x\mapsto\sin\frac{1}{x} is integrable on (0,1] as well).
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    If it is integrable, then what is the integral over (0,1]??
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    Quote Originally Posted by Amanda1990 View Post
    If it is integrable, then what is the integral over (0,1]??
    This is not an easy integral to compute...

    By letting u=\frac{1}{x}, you get \int_0^1\sin\frac{1}{x}dx=\int_1^\infty \frac{\sin u}{u^2} du

    I don't think there is a simple formula for this one; does anyone know?
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  5. #5
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    Quote Originally Posted by Laurent View Post
    This is not an easy integral to compute...

    By letting u=\frac{1}{x}, you get \int_0^1\sin\frac{1}{x}dx=\int_1^\infty \frac{\sin u}{u^2} du

    I don't think there is a simple formula for this one; does anyone know?
    Integration by parts gives:

    \int \sin(1/x)\ dx = x \sin\left(\frac{1}{x}\right)-\text{Ci}\left(\frac{1}{x}\right) + C

    So:

    \int_0^1 \sin(1/x)\ dx = \sin(1)-\text{Ci}(1)-\lim_{a \to 0}\left[ a\sin\left(\frac{1}{a}\right)-\text{Ci}\left(\frac{1}{a}\right)\right] =\sin(1)-\text{Ci}(1)\approx 0.504067

    CB
    Last edited by CaptainBlack; February 10th 2009 at 06:36 AM. Reason: added constant of integration
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    Ok thanks very much for this. I see now that I was being a bit silly when I suggested that sin(1/x) was probably not measurable.

    Perhaps this is a better question to highlight the problems I'm having with Lebesgue integration:

    Is f(x) tan x integrable over the interval (0,pi/2)?

    Now we can show that f is measurable on the interval, and hence continuous. However, it is unbounded and hence not necessarily integrable. I know we can say " \int tan x = \log \sec x " and probably this to show the integral doesn't exist, but this isn't really the point. I'm after something that show from first principals or standard results (eg Monotone convergence theorem etc) why this isn't integrable over the stated interval. Thanks.
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