Engelbert–Schmidt zero–one law

Let ${\displaystyle {\mathcal {F}}}$ be a σ-algebra and let ${\displaystyle F=({\mathcal {F}}_{t})_{t\geq 0}}$ be an increasing family of sub-σ-algebras of ${\displaystyle {\mathcal {F}}}$. Let ${\displaystyle (W,F)}$ be a Wiener process on the probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$. Suppose that ${\displaystyle f}$ is a Borel measurable function of the real line into [0,∞]. Then the following three assertions are equivalent:

(i) ${\displaystyle P\left(\int _{0}^{t}f(W_{s})\,ds<\infty ,{\text{ for all }}t\geq 0\right)>0}$

(ii) ${\displaystyle P\left(\int _{0}^{t}f(W_{s})\,ds<\infty {\text{ for all }}t\geq 0\right)=1}$

(iii) ${\displaystyle \int _{K}f(y)\,dy<\infty \,}$

for all compact subsets ${\displaystyle K}$ of the real line.[4]

• zero-one law