Is the Pre-Image of a Measurable Function on a Measure Space also Measurable?

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In summary, a measurable function is a mathematical function that maps elements of a measurable space to real numbers, preserving the structure of the space. The pre-image of a measurable function is the set of elements in the domain that map to a given set of real numbers in the range. However, the pre-image is not always measurable and depends on the function and measure space. For example, the function f(x) = 0 if x is rational and f(x) = 1 if x is irrational has a non-measurable pre-image. The measurability of the pre-image is important because it allows for the definition and manipulation of integrals, and aids in studying the properties of measurable functions and establishing theorems in measure theory.
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Chris L T521
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MHB
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Here's this week's problem!

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Problem: Let $f$ be a measurable function on a measure space $(X,\Lambda,\mu)$. Show that the pre-image of any Borel set of $\mathbb{R}$ is also in $\Lambda$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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I will edit this post soon (by this afternoon PST) and update it with the solution then (since it's still being decided upon due to different interpretations).
 

FAQ: Is the Pre-Image of a Measurable Function on a Measure Space also Measurable?

1. What is a measurable function?

A measurable function is a mathematical function that maps elements of a measurable space to real numbers. In other words, it is a function that preserves the structure of a measurable space.

2. What is a pre-image of a measurable function?

The pre-image of a measurable function is the set of all elements in the domain that map to a given set of real numbers in the range of the function.

3. Is the pre-image of a measurable function always measurable?

No, the pre-image of a measurable function is not always measurable. It depends on the function and the measure space it is defined on.

4. Can you provide an example of a measurable function with a non-measurable pre-image?

Yes, consider the function f: [0,1] → [0,1] defined as f(x) = 0 if x is rational and f(x) = 1 if x is irrational. The pre-image of the set {0} under this function is the set of all rational numbers, which is not measurable.

5. Why is the measurability of the pre-image of a measurable function important?

The measurability of the pre-image of a measurable function is important because it allows us to define and manipulate integrals of measurable functions. It also helps us to study the properties of measurable functions and establish important theorems in measure theory.

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