  The action and averaging properties of conditional expectation operators are
studied in the, measure-free, Riesz space, setting of Kuo, Labuschagne and
Watson [{Conditional expectations on Riesz spaces}, J. Math. Anal. Appl., 303
(2005), 509-521] but on the abstract $L^2$ space, ${\cal L}^2(T)$ introduced by
Labuschagne and Watson [{ Discrete Stochastic Integration in Riesz Spaces},
Positivity, 14, (2010), 859 - 575]. In this setting it is shown that
conditional expectation operators leave ${\cal L}^2(T)$ invariant and the
Bienaym\'e equality and Tchebichev inequality are proved.
  From this foundation Bernoulli processes are considered. Bernoulli's strong
law of large numbers and Poisson's theorem are formulated and proved.
