Conditions on f for dx/f(x) to be a Measure in L^2

[Jip]

Hi,
Let's say I consider the real numbers and some function real function f, nowhere zero, and positive.
My question is, what are the conditions on f for dx/f(x) to be a valid measure on this space?

(I have to consider a Hilbert space L^2(R, dx/f(x)) with scalar product a.b = \int a^*(x) b(x) \frac{dx}{f(x)}

I'm a physicist, so please excuse me if this is not written in perfect mathematical language!
Thanks

 

[wabbit]

To get a valid measure you just need it (1/f) (to be measurable and) to sum to one; but your Hilbert space will contain only functions a such that the integral of a*a/f is finite. If you need this to apply to a specific class of functions then you'll have to restricf f accordingly.

 

[Hawkeye18]

In addition to you assumptions (everywhere or almost everywhere positive) it needs to be measurable, and that is all.

 

[Jip]

In addition to you assumptions (everywhere or almost everywhere positive) it needs to be measurable, and that is all.

Thanks! And when 1/f is measurable, is it also true for f?

 

[Hawkeye18]

Yes, $1/f$ is measurable if and only if $f$ is.