But if a random variable is 0 with probability measure 1 and is undefined with probability measure 0, we can’t just say it’s identical to the zero random variable or that it has expected value zero (I think, happy to be corrected with a link to a math source).
The definition of expected value is ∫xP[x]. If the set of discontinuities of a function has measure zero, then it is still Riemann integrable. So the integral exists despite not being identical to the zero random variable, and the value is zero. In the general case you have to use measure theory, but I don’t think it’s needed here.
Also, there’s no reason our intuitions about the goodness of the infinite sequence of bets has to match the expected value.
The definition of expected value is ∫xP[x]. If the set of discontinuities of a function has measure zero, then it is still Riemann integrable. So the integral exists despite not being identical to the zero random variable, and the value is zero. In the general case you have to use measure theory, but I don’t think it’s needed here.
Also, there’s no reason our intuitions about the goodness of the infinite sequence of bets has to match the expected value.