Real Analysis Chapter 2 Solutions

1. Suppose f is measurable. Then f−1({−∞}) ∈ M and f−1({∞}) ∈ M, because {−∞} and {∞} are Borel sets. If B ⊆ R is Borel then f−1(B) ∈M, and hence f−1(B) ∩ Y ∈M (since R is also Borel). Thus f is measurable on Y. Conversely, suppose that f−1({−∞}) ∈ M, f−1({∞}) ∈ M and f is measurable on Y. Let B ⊆ R be Borel. Then f−1(B) ∩ Y ∈ M, and f−1(B) = (f−1(B) ∩ Y ) ∪ (f−1(B) \ Y ). Clearly f−1(B) \ Y = f−1(B ∩ {−∞,∞}), which is measurable because it is either ∅, f−1({−∞}), f−1({−∞}) or f−1({−∞})∪f−1({−∞}). This implies that f−1(B) ∈ M, so f is measurable.