A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω,
, ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] by
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410006494X/resource/name/S030500410006494X_eqn001.gif?pub-status=live)
for A ∈
and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation if
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410006494X/resource/name/S030500410006494X_eqn002.gif?pub-status=live)
is finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’