I thought I'd jump in as the author of the original post.
The context is that I'm trying to write an introduction to Bayesian stats for people who know calc and matrices, but may not have taken or understood math stats. Specifically, I want to (a) use the notation that's commonly used in the field (e.g. in Andrew Gelman et al.'s books, Michael Jordan et al.'s papers, etc.), and (b) not confuse readers with a long introduction to sample spaces and a sketchy description of measures, just so I could introduce precise random variable notation only to abuse it.
The big problem with trying to define continuous densities is you never get enough measure theory in an intro to probability (e.g. DeGroot and Schervis, Larsen and Marx) to bottom out in a real definition. It's not that complex, so if you're interested, I'd highly recommend Kolmogorov's own intro to analysis, which has great coverage of both Lebesgue integration (so you can understand the usual R^n case) and general measure theory (so you can impress your friends with your knowledge of analysis).
The context is that I'm trying to write an introduction to Bayesian stats for people who know calc and matrices, but may not have taken or understood math stats. Specifically, I want to (a) use the notation that's commonly used in the field (e.g. in Andrew Gelman et al.'s books, Michael Jordan et al.'s papers, etc.), and (b) not confuse readers with a long introduction to sample spaces and a sketchy description of measures, just so I could introduce precise random variable notation only to abuse it.
The big problem with trying to define continuous densities is you never get enough measure theory in an intro to probability (e.g. DeGroot and Schervis, Larsen and Marx) to bottom out in a real definition. It's not that complex, so if you're interested, I'd highly recommend Kolmogorov's own intro to analysis, which has great coverage of both Lebesgue integration (so you can understand the usual R^n case) and general measure theory (so you can impress your friends with your knowledge of analysis).