Componentwise absolute continuity

Question

It is clear that if $\mathbf{X}=(X_1,\dots,X_n)$ is an absolutely continuous random vector, then $X_i$ is absolutely continuous for all $i$.

Is the converse true?

What I know

• $\mathbf{X}$ is absolutely continuous if and only if $P(\mathbf{X}\in A)=0$ for all borel set $A$ of measure $0$.
• $\mathbf{X}$ is absolutely continuous if and only if there is a function $f$ such that $P(\mathbf{X}\in A)=\int_{A}f$.
• It's obvious if $\{X_i\}_{i=1}^n$ are independent.

Any idea/reference for a proof of this result?

Answer 1

The converse is absolutely not true. Let $X$ be any absolutely continuous random variable. Then the random vector $(X,X)$ is not absolutely continuous, even though its components are. Indeed, letting $A$ be the set of points on the lines $y=x$, then $P\big((X,X)\in A\big )=1$, even though $A$ is a set with Lebesgue measure $0$.