Let be a σ-algebra and let be an increasing family of sub-σ-algebras of . Let be a Wiener process on the probability space .
Suppose that is a Borel measurable function of the real line into [0,∞].
Then the following three assertions are equivalent:
(i) .
(ii) .
(iii) for all compact subsets of the real line.[4]
In 1997 Pio Andrea Zanzotto proved the following extension of the Engelbert–Schmidt zero-one law. It contains Engelbert and Schmidt's result as a special case, since the Wiener process is a real-valued stable process of index .
Let be a -valued stable process of index on the filtered probability space .
Suppose that is a Borel measurable function.
Then the following three assertions are equivalent:
(i) .
(ii) .
(iii) for all compact subsets of the real line.[5]
The proof of Zanzotto's result is almost identical to that of the Engelbert–Schmidt zero-one law. The key object in the proof is the local time process associated with stable processes of index , which is known to be jointly continuous.[6]