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Consider a random vector $(X,Y)$ with distribution function $F:\mathbb{R}^2\rightarrow [0,1]$ and let $\mu_F$ be the associated measure.

Take any $(a,b)\in \mathbb{R}^2$ and consider the set $$ \mathcal{R}_{(a,b)}\equiv \{(x,y)\in \mathbb{R}^2: x+a=y+b\}. $$

Is assuming that $F$ is a continuous distribution (in the sense outlined here) sufficient to ensure that $$ \mu_F(\mathcal{R}_{(a,b)})=0 \text{ }\forall (a,b)\in \mathbb{R}^2 $$ or do we need absolute continuity (or, other conditions?)

TEX
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    I believe you can reduce the problem by a dimension, making it instantly answerable: simply consider the random variable $X-Y.$ – whuber Oct 06 '20 at 17:32
  • Thanks. Therefore, I think I need absolute continuity, correct? – TEX Oct 06 '20 at 17:42
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    Absolute continuity would be sufficient. It's not a necessary condition. Any necessary condition would be equivalent to $X-Y$ having an absolutely continuous distribution. There are distributions on the plane that are not absolutely continuous yet have one absolutely continuous marginal (and you can think of the distribution of $X-Y$ as a marginal after rotating the picture by 45 degrees). – whuber Oct 06 '20 at 17:44

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