Discussion Overview
The discussion revolves around the nature of probability mass functions (pmf) and probability density functions (pdf) as random variables within the context of measurable functions. Participants explore whether these functions can be considered measurable when defined over different types of domains, particularly focusing on the implications of countable versus uncountable domains.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that a pdf is always measurable and integrable, as it is defined to be integrable.
- Others argue that a pmf is measurable since its domain is countable, but question its measurability when defined over an uncountable domain like the real numbers.
- A participant suggests that a pmf remains measurable because most numbers in an uncountable domain are sent to zero, leaving only a countable number of non-zero elements, which are interesting.
- Another participant elaborates on the measurability of a pmf by discussing the countable range and the properties of the Borel algebra, asserting that the pre-images of subsets are also elements of the Borel algebra.
- There is a challenge regarding the definition of pmf on the real numbers, with references to external sources to support claims.
- One participant acknowledges a misunderstanding regarding the discussion of pmf versus random variables.
Areas of Agreement / Disagreement
Participants express differing views on the measurability of pmf when defined over uncountable domains, indicating that the discussion remains unresolved with multiple competing perspectives.
Contextual Notes
The discussion includes assumptions about the definitions of pmf and pdf, the nature of measurable functions, and the implications of countability in probability theory, which are not fully resolved.
