A Measurable Function not Borel Measurable

Thread starter zling110
Start date Mar 15, 2009
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Function Measurable

In summary, a measurable function is a mathematical function that preserves certain properties, such as continuity or measurability, while a Borel measurable function preserves the Borel structure of the input and output spaces. Measurable functions are more general than Borel measurable functions, as there are measurable functions that are not Borel measurable. An example of a measurable function that is not Borel measurable is the indicator function for the Vitali set. Distinguishing between these two types of functions is important in understanding and working with different levels of structure in mathematical spaces, and can be useful in solving problems in various fields of mathematics.

Mar 15, 2009

zling110

Most books have the famous example for a Lebesgue measurable set that is not Borel measurable. However, I am trying to find a Lebesgue measurable function that is not Borel measurable. Can anyone think of an example?

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Mar 15, 2009

dvs

Use the characteristic function of a set that's Lebesgue but not Borel.

Mar 15, 2009

zling110

dvs said:

Use the characteristic function of a set that's Lebesgue but not Borel.

thanks

1. What is a measurable function?

A measurable function is a mathematical function that maps inputs to outputs in a way that preserves certain properties, such as continuity or measurability. In other words, the function is defined in such a way that the pre-images of measurable sets are also measurable.

2. What does it mean for a function to be Borel measurable?

A Borel measurable function is a function that maps inputs to outputs in a way that preserves the Borel structure of the input and output spaces. This means that the pre-images of Borel sets are also Borel sets.

3. How is a measurable function different from a Borel measurable function?

A measurable function is a more general concept than a Borel measurable function. While all Borel measurable functions are also measurable, there are measurable functions that are not Borel measurable. This means that there are functions that preserve measurability but do not necessarily preserve the Borel structure of the input and output spaces.

4. What is an example of a measurable function that is not Borel measurable?

An example of a measurable function that is not Borel measurable is the indicator function for the Vitali set. This function maps real numbers to 0 or 1, depending on whether the number is a member of the Vitali set or not. While this function preserves measurability, it does not preserve the Borel structure since the Vitali set is not a Borel set.

5. Why is it important to distinguish between measurable and Borel measurable functions?

Distinguishing between measurable and Borel measurable functions is important because it allows us to understand and work with different levels of structure in mathematical spaces. It also helps us to identify which functions preserve certain properties and which do not, and can be useful in proving theorems and solving problems in various fields of mathematics, such as measure theory and topology.