4. Measurability - Probability Tutorials

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Tutorial 4: Measurability14. MeasurabilityDefinition 25LetAandBbe two sets, andf:A Bbe a A,wecalldirect imageofA byfthe set denotedf(A ),anddefinedbyf(A )={f(x):x A }.Definition 26LetAandBbe two sets, andf:A Bbe a B,wecallinverse imageofB byfthe set denotedf 1(B ), and defined byf 1(B )={x:x A, f(x) B }.Exercise two sets, andf:A Bbe a AandB Explain why the notationf 1(B ) is potentially Show that the inverse image ofB byfis in fact equal to thedirect image ofB byf Show that the direct image ofA byfis in fact equal to theinverse image ofA byf 4: Measurability2Definition 27Let( ,T)and(S,TS)be two topological spaces. Amapf: Sis said to becontinuousif and only if: B TS,f 1(B) TIn other words, if and only if the inverse image of any open set inSis an open set in.

Tutorial 4: Measurability 6 Theorem 12 Let (E,d) be a metric space and F ⊆ E.Then,the topology on F induced by the metric topology, is equal to the metric topology on F associated with the induced metric…

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