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zling110
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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?
dvs said:Use the characteristic function of a set that's Lebesgue but not Borel.
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.
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.
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.
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.
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.