1. Let f be a measurable function from X onto S where (X, A) is a measurable space and (S, e) is a metric space with Borel σ-algebra. Let T be a subset of S with discrete relative topology (all subsets of T are open in T ). Show that there is a measurable function g from X onto T . Hint: For f (x ) close enough to t ∈ T , let g(x ) = t ; otherwise, let g(x ) = to for a fixed to ∈ T .
2. Let f be a Borel measurable function from a separable metric space X onto a metric space S with metric e. Show that (S, e) is separable. Hints: As in Problem 8, X has at most c Borel sets. If S is not separable, then show that for some ε> 0 there is an uncountable subset T of S with d(y, z) >ε for all y /= z in T . Use Problem 9 to get a measurable function g from X onto T . All g-1( A), A ⊂ T , are Borel sets in X .