Real- Valued Random Variable  This is from Ross Ihaka's notes about Time Series Analysis. 

Any random variable which has probability 1 of being zero will have $\langle X,X \rangle = 0$, which violates the requirement that this only happen when $X=0$.



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*How can a real-valued random variable has probability 1 of being zero and probability 0 of being other values? 

*Why is there uniqueness problem here?


Thank you.

A random variable is a (measurable) function from a sample space to the real numbers. There is nothing wrong with mapping (almost) everything to value 0. This is still a random variable.
When $X=0$ with probability $1$, $\langle X,X\rangle$ equals to 0.
Here is an example, let the sample space be the interval $[0,1]$ with uniform measure. For a $x$ in the sample space, $X(x) = 0$ for all $x$ in the sample space except $X(0.5)=1$, then $EX^2=0$ but $X$ is not 0 everywhere. This violates uniqueness that there is only one $X$ such that $EX^2=0$
This is quite standard in probability theory: you just treat two r.v. as the same if they equal almost everywhere, i.e. they differ only on a set of measure $0$. You 'quotient' out the equivalent classes of random variables which are the same almost everywhere, then the 0 is unique. This is known as $L^2$ space.
from a practical point of view, just ignore it, it does not matter.
