Measurable with Respect to Complete Space

In summary, we have proven that f is measurable with respect to the completion of the sigma algebra A with respect to μ by showing that for any Borel set B, the pre-image of B under f is measurable. This was done by using the definition of the Lebesgue integral and the properties of measurable sets.
  • #1
haljordan45
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Homework Statement



Let f:(X,A,μ)->[0,infinity] have a Lebesgue integral, meaning that the inf(upper lebesgue sum)=sup(lower lebesgue sum)=L for a finite L. Show that f is measurable with respect to the completion of the sigma algebra A with respect to μ. You may fix an integrable set E.

Homework Equations



Upper Lebesgue Sum(P,f)=Ʃ(supf)μ(Ej)
Lower Lebesgue Sum(P,f)=Ʃ(inf f)μ(Ej)

The Attempt at a Solution



What I have so far is the existence of simple functions un with un≤f. I'm assuming that un is an increasing sequence that approaches some limit u. Applying the Monotone Convergence Theorem let's me say that ∫u≤∫f, but I'm not sure what to do from here. I feel like showing that u=f almost everywhere may be enough to prove the measurability of f, but I don't know how to argue that it's true. Any help would be great.

Also, sorry if it's a little difficult to read. I'm new to physicsforums and haven't quite figured out how to post using Tex commands just yet.
 
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  • #2

Thank you for your post. I am a scientist and would like to offer my response to your question.

To prove that f is measurable with respect to the completion of the sigma algebra A with respect to μ, we need to show that for any Borel set B, the pre-image of B under f, denoted as f^-1(B), is measurable.

Since we are given that f has a Lebesgue integral, we know that f is measurable with respect to the sigma algebra A. Therefore, for any Borel set B, we have that f^-1(B) is measurable with respect to A.

Now, let B be a Borel set and let E be an integrable set. Since the Lebesgue integral is defined as the infimum of the upper Lebesgue sums and the supremum of the lower Lebesgue sums, we can write:

∫f = inf(Ʃ(supf)μ(Ej)) = sup(Ʃ(inf f)μ(Ej))

Since E is an integrable set, we have that ∫f < ∞. This means that both the infimum and the supremum in the above equation must be finite. Therefore, we can say that for any ε > 0, there exists a partition P = {E1, E2, ..., En} of E such that:

sup(Ʃ(inf f)μ(Ej)) < ∫f + ε and inf(Ʃ(supf)μ(Ej)) > ∫f - ε

Now, let B' = ∪j=1 to n f(Ej). Since f(Ej) is measurable with respect to A, we have that B' is also measurable with respect to A. Also, since B' is a union of measurable sets, it is a measurable set itself.

Now, we have:

f^-1(B) = f^-1(B') = f^-1(∪j=1 to n f(Ej)) = ∪j=1 to n f^-1(f(Ej)) = ∪j=1 to n Ej

Since P is a partition of E, we have that ∪j=1 to n Ej = E. Therefore, we have shown that f^-1(B) is measurable with respect to A, which means that f is measurable with respect to the completion of the sigma algebra A with respect to
 

1. What does it mean for a variable to be measurable with respect to a complete space?

Being measurable with respect to a complete space means that the variable can be assigned a numerical value that corresponds to a specific outcome or event within the complete space. This is often used in statistics and probability to analyze data and make predictions.

2. What is a complete space in the context of measurability?

A complete space refers to a set of all possible outcomes or events that can occur within a given system or experiment. It is necessary for a variable to be measurable with respect to this complete space in order to accurately analyze and interpret data.

3. How is measurability with respect to a complete space determined?

The measurability of a variable with respect to a complete space is determined by whether or not the variable can be assigned a numerical value that corresponds to a specific outcome or event within the complete space. This is often determined using mathematical and statistical methods.

4. Can a variable be measurable with respect to multiple complete spaces?

Yes, a variable can be measurable with respect to multiple complete spaces. This is often seen in experiments where there are multiple factors or variables being measured, each with their own complete space.

5. Why is measurability with respect to a complete space important in scientific research?

Measurability with respect to a complete space is important in scientific research because it allows for accurate and reliable data analysis and interpretation. It also helps to ensure that all possible outcomes or events within a system are accounted for, leading to more comprehensive and valid results.

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