 A classical probability measure space is a triplet \((X,{\cal S},p)\), where X is the set of elementary random events, \({\cal S}\) is a Boolean algebra of some subsets of X and p is the additive probability measure from \({\cal S}\) into the unit interval \([0,1]\). The probability measure space \((X,{\cal S},p)\) is called common cause complete if there exists a common cause C in \({\cal S}\) of every correlated pair \((A,B)\) of elements \(A,B\) in \({\cal S}\), common cause incomplete otherwise. One can show that common cause (in)completeness can be characterized in terms of the measure theoretic (non)atomicity of probability measure spaces: \((X,{\cal S},p)\) is common cause complete if and only if \((X,{\cal S},p)\) contains at most one measure theoretic atom (Gyenis &amp; Rédei 2011). A measure theoretic atom is an element \(A_0\) in \({\cal S}\) such that \(p(A_0)\not=0\), and, if \(B\subset A_0\), then \(p(B)=0\). In particular measure theoretically purely non-atomic probability measure spaces (i.e., probability spaces that do not contain any measure theoretic atom) are common cause complete (Gyenis &amp; Rédei 2011; Hofer-Szabó et al. 2013; Marczyk &amp; Wroński 2015). An example of a measure theoretically purely non-atomic probability measure space is the unit interval \([0,1]\) with the uniform probability (Lebesgue measure) on the Lebesgue measurable subsets of \([0,1]\); a class of examples are probability measures on the (Lebesgue measurable subsets of the) real line where the probability is given by a density function with respect to the Lebesgue measure on the real line.