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

In summary, the conversation discusses the conditions for dx/f(x) to be a valid measure in a Hilbert space L^2(R, dx/f(x)). It is mentioned that for it to be a valid measure, 1/f must be measurable and sum to one. In addition, the function f must be measurable and positive everywhere or almost everywhere. The conversation also mentions that if 1/f is measurable, then f is also measurable.
  • #1
Jip
20
2
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 [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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  • #2
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.
 
  • #3
In addition to you assumptions (everywhere or almost everywhere positive) it needs to be measurable, and that is all.
 
  • #4
Hawkeye18 said:
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?
 
  • #5
Yes, ##1/f## is measurable if and only if ##f## is.
 

1. What is the definition of "conditions on f for dx/f(x) to be a measure in L^2"?

The conditions on f for dx/f(x) to be a measure in L^2 refer to the requirements that must be met for the function f to be considered a measure in the L^2 space, which is a function space commonly used in mathematics and physics.

2. Why is it important for dx/f(x) to be a measure in L^2?

It is important for dx/f(x) to be a measure in L^2 because it allows for the use of various mathematical tools and techniques in the L^2 space, which can greatly aid in the analysis and understanding of various physical phenomena.

3. What are the main conditions that must be satisfied for dx/f(x) to be a measure in L^2?

The main conditions for dx/f(x) to be a measure in L^2 include f being a bounded function, f being continuous almost everywhere, and the integral of f over the entire domain being finite.

4. How does the concept of "measurability" play a role in dx/f(x) being a measure in L^2?

In order for dx/f(x) to be a measure in L^2, f must be a measurable function. This means that the preimage of any Borel set under f must be a measurable set, which ensures that the measure defined by dx/f(x) is well-defined and consistent.

5. Can dx/f(x) still be a measure in L^2 if f does not satisfy all the necessary conditions?

No, dx/f(x) cannot be considered a measure in L^2 if f does not meet all the required conditions. These conditions are essential for ensuring that the measure is well-defined and can be used effectively in mathematical and physical applications.

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