SUMMARY
The discussion focuses on identifying a function within a measurable space that is non-measurable while ensuring that both its absolute value, |f|, and its square, f², are measurable. The key conclusion is that for any subset S of the real numbers, the preimage under |f| and f² results in either the entire space X or the empty set, depending on whether S includes the value 1. This establishes that |f| and f² maintain measurability despite f being non-measurable.
PREREQUISITES
- Understanding of measurable spaces and functions
- Familiarity with the concepts of absolute value and squaring functions
- Knowledge of preimages in the context of set theory
- Basic comprehension of real analysis
NEXT STEPS
- Study the properties of measurable functions in real analysis
- Explore the concept of non-measurable sets and functions
- Learn about the implications of preimages in measurable spaces
- Investigate examples of measurable and non-measurable functions
USEFUL FOR
Mathematicians, students of real analysis, and anyone interested in the properties of measurable and non-measurable functions.