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


1 Answer 1


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 $\Bbb R^n$

  • $\begingroup$ So, for example, a countable set of planes in 3 dimensions, and hyperplanes in higher dimensions? $\endgroup$ Feb 19, 2018 at 1:57
  • $\begingroup$ Yes that 's true because rather that a plane may have a non zero measure in 2 dimensions it has volume zero and the measure in a 3 dimensions space is the volume. $\endgroup$ Feb 19, 2018 at 8:48

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