# Conditional independence: conditioning on an empty set of random variables

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)?

• What is the measure of an empty set? – Alecos Papadopoulos Oct 12 '14 at 16:00
• 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 $\{\varnothing,\Omega\}$. – whuber Oct 21 '14 at 17:43

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.