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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 [tex] L^2(R, dx/f(x)) [/tex] with scalar product [tex]a.b = \int a^*(x) b(x) \frac{dx}{f(x)} [/tex]

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

Thanks

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# Measured space

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