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Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is always a conditional independence (but the converse is false)?

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    $\begingroup$ What is the measure of an empty set? $\endgroup$ – Alecos Papadopoulos Oct 12 '14 at 16:00
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    $\begingroup$ It might help to recognize that the sigma algebra generated by a set of random variables is the smallest sigma algebra making them all measurable. Since all sigma algebras over a set $\Omega$ must, by definition, include $\varnothing$ and $\Omega$, then your question comes down to understanding conditioning on the sigma algebra $\{\varnothing,\Omega\}$. $\endgroup$ – whuber Oct 21 '14 at 17:43
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Yes, I would say, $X \perp\!\!\!\perp Y$ can be thought of as conditional independence, but conditioning on the full sample space $\Omega$. The probability of the empty set impossible event is zero, and you cannot really condition on an event with probability zero.

Conditioning on the full sample space really is saying that "the only fact that I know is that some outcome in the sample space happened", as it must if your model is any good.

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