Discussion Overview
The discussion centers on the relationship between Riemann integrability and Lebesgue integrability, specifically whether a function that is not Riemann integrable must also be not Lebesgue integrable. Participants explore examples, counterexamples, and conditions under which these integrability concepts apply.
Discussion Character
- Debate/contested
- Exploratory
- Mathematical reasoning
Main Points Raised
- Some participants assert that a function not being Riemann integrable does not imply it is also not Lebesgue integrable, citing examples like the characteristic function of the rationals.
- Another example provided is a function defined as 1 at rational points of the form 1/N and 0 elsewhere, which is not Riemann integrable but Lebesgue integrable.
- Participants discuss conditions under which a function that is continuous almost everywhere may still be Lebesgue integrable despite being non-Riemann integrable.
- Concerns are raised about unbounded functions, such as f(x) = 1/x on (0,1], which are not Riemann integrable and are suggested to be intuitively not Lebesgue integrable.
- Some participants mention that the integral of sin(x)/x over an infinite range is Riemann integrable but not Lebesgue integrable, highlighting the differences in definitions and applicability of integrals over infinite intervals.
- There is a discussion about the nature of Riemann integrals over infinite intervals and their relationship to Lebesgue integrals, with some participants questioning the validity of such integrals.
- Clarifications are made regarding the definitions of measurable functions and their implications for Riemann and Lebesgue integrability.
Areas of Agreement / Disagreement
Participants generally disagree on the implications of non-Riemann integrability for Lebesgue integrability, with multiple competing views and examples presented. The discussion remains unresolved regarding the broader implications and specific cases.
Contextual Notes
Some participants note that the relationship between Riemann and Lebesgue integrability may depend on the boundedness of functions and the nature of the intervals considered. There are also discussions about the definitions of measurable functions and their relevance to integrability.