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One integral inequality

  1. Mar 8, 2008 #1
    Let X be a measure space, and [itex]f:X\times X\to [0,\infty[[/itex] some integrable function. Is the following inequality always true,

    [tex]
    \int\limits_{X} dx\;f(x,x)\; \leq\; \sup_{x_1\in X} \int\limits_{X} dx_2\; f(x_1,x_2) ?
    [/tex]
     
    Last edited: Mar 8, 2008
  2. jcsd
  3. Mar 8, 2008 #2

    mathman

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    Gold Member

    No. Let f(x,y)=sinxsiny for 0<=x,y,<=2pi and zeo otherwise. The left integral is pi, while the right integral is 0.
     
  4. Mar 8, 2008 #3
    I see.

    [tex]
    f:[0,2\pi]\times [0,2\pi]\to [0,\infty[,\quad f(x,y) = \sin(x)\sin(y) + 1
    [/tex]

    gives a counter example that satisfies the original conditions.
     
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