Notes on Sigma Algebras for Brownian Motion Course

The space R comes with its standard topology which we denote byO (that is, O is the collection of open sets in R). It comes also with its Borel sigma algebra which we denote by B. The Borel sigma algebra is the smallest one containing all open sets (i.e., the sigma algebra generated by O). Let I be an arbitrary non-empty set (finite, countable or uncountable). For x ∈ I, the coordinate function Tx is the function Tx : RI → R defined by Tx(f) := f(x). The set RI is naturally endowed with both a topology and a sigma algebra named, naturally, the product topology and product sigma algebra. The product topology OI is the smallest topology making all the (Tx), x ∈ I, continuous. The product sigma algebra BI is the smallest sigma algebra making all the (Tx), x ∈ I, measurable (with respect to the Borel sigma algebra on R). Slightly more explicitly, a base for the product topology is given by all open cylinder sets which are the sets of the form ∏