In: Advanced Math
Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X.
(a) What does it mean for a function f : X → R to be measurable? [2%] If f and g are measurable, show that the function f − g is also measurable. [6%]
(b) Let (fn) be a sequence of measurable functions.
(i) What does it mean to say that (fn) converges pointwise to a function f? [2%]
(ii) If (fn) converges pointwise to f, show that f is a measurable function.
Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X.
(a) What does it mean for a function f : X → R to be measurable? [2%] If f and g are measurable, show that the function f − g is also measurable. [6%]
(b) Let (fn) be a sequence of measurable functions.
(i) What does it mean to say that (fn) converges pointwise to a function f? [2%]
(ii) If (fn) converges pointwise to f, show that f is a measurable function.