What is a "strictly positive distribution"? I am reading Judea Pearl's "Causality" (second edition 2009) and in section 1.1.5 Conditional Independence and Graphoids, he states:

The following is a (partial) list of properties satisfied by the 
  conditional independence relation (X_||_Y | Z). 
  
  
*
  
*Symmetry: (X_||_ Y | Z) ==> (Y_||_X | Z).
  
*Decomposition: (X_||_ YW | Z)  ==> (X_||_Y | Z).
  
*Weak union: (X_||_ YW | Z)  ==> (X_||_Y | ZW). 
  
*Contraction: (X_||_ Y | Z) & (X_||_ W | ZY) ==> (X_||_ YW | Z).
  
*Intersection: (X_||_  W | ZY) & (X_||_  Y | ZW) (X_||_  YW | Z). 
  
  
  (Intersection is valid in strictly positive probability distributions.)
(formula (1.28) given earlier in the publicatiob: [(X_||_ Y | Z) iff P(X | Y,Z ) = P(X | Z)  )

But what is an  "strictly positive distribution" in general terms, and what distinguishes a "strictly positive distribution" form a distribution that is not strictly positive?
 A: A strictly positive distribution $D_{sp}$ has values $D_{sp}(x)>0$ for all $x$. This is different from a non-negative distribution $D_{nn}$ where $D_{nn}(x) \geq 0$.
A: The mass of each ball bearing in a population of ball bearings would be strictly positive because something with zero mass cannot be a ball bearing.
A: A strictly positive probability distribution over a state space simply means that all states are possible, ie no state has a probability of zero. All states have a probability greater than zero. "Strictly positive" means greater than zero.
Strictly positive does not imply that the probability of any state could be negative. There is no such thing as negative probability.
A: As an example illustrating the definition of a strictly positive probability distribution in action (Courtesy of an old paper by Richard Holley on FKG Inequalities), imagine that we have $\Lambda$ which is a finite fixed set. Imagine also that we have $\Gamma$, which is a sublattice of the lattice of subsets of $\Lambda$. Let us then let $\mu$ be a strictly positive probability distribution on some finite distributed lattice $\Gamma$. For $\mu$ to be strictly positive, $\mu(A)>0$ for all $A\in\Gamma$ and $\sum_{A\in\Gamma}\mu(A)=1$
