One integral inequality

  • Thread starter jostpuur
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  • #1
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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]
 
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Answers and Replies

  • #2
mathman
Science Advisor
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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.
 
  • #3
2,112
18
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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