TITLE: question on sigma-fields
QUESTION [5 upvotes]: Let X,Y to be mappings from the sample space Ω to R and suppose Y is measurable with respect to σ(X), the smallest σ-field that makes X measurable. 
Does it follow that there exists some Borel-measurable function f: R → R such that 
Y=f(X)?

REPLY [2 votes]: It is trivially true but maybe worth noting that the converse is also true - if there exists such an f, then Y is σ(X)-measurable.
This and the question asked are theorem 20.1(ii) in Billingsley's Probability and Measure, 3rd edition.