In one dimension, it is true that the integral of a function over a point (like from $a$ to $a$) is 0, and in general, changing the value of the function at a finite number of points has no effect on the value of the integral.As a result, the probability of $x=5$ for a continuous distribution is 0
My question is, what is the condition in $\mathbb{R}^m$? My hypothesis is that the event is of 0 zero in the appropriate measure for that dimension (ie, no length for 1 dimension, no area for 2 dimensions, no volume for 3 dimensions), but I cannot find this written down.
For example, in $\mathbb{R}^2$, would it be sufficient for the event to have no area? For example, it seems like intuitively, $P(X=2Y)$ should be 0. Any sort of reference to stat book or math book that proves some condition like this would be much appreciated
The condition is to change the value of the probability distribution function on a zero set. A zero set in a metric space $X$ is a subset of $X$ with Lebesgue measure zero. In $\Bbb R$ a zero set can be any countable set of points or Cantor sets. In $\Bbb R^2$ it can be any countable set of lines and so on for $\Bbb R^n$