L^p spaces are not equivalent for infinite measure sets

[JasonJo]

Find a function f such that f is in L^P(R) but not in L^Q(R) for p not equal to q, where R is the set of real numers.

I'm guessing I need to find a function that only blows up when it is raised to the qth power, but I am having some difficulty proving this.

 

[morphism]

Try finding some function f that's in L^1 but not in L^2. To make things easier, you can choose a compactly-supported f.

 

[quasar987]

I suppose just f = sin would do!

 

[morphism]

It wouldn't, because sin isn't in any L^p(R) space.

 

[quasar987]

Oh, right!