Are Integral Inequalities Always True in Measure Spaces?

[jostpuur]

Let X be a measure space, and $f:X\times X\to [0,\infty[$ some integrable function. Is the following inequality always true,

$$\int\limits_{X} dx\;f(x,x)\; \leq\; \sup_{x_1\in X} \int\limits_{X} dx_2\; f(x_1,x_2) ?$$

 

[mathman]

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.

 

[jostpuur]

I see.

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

gives a counter example that satisfies the original conditions.