Measure, Integration & Real Analysis · Problem 2B.15

Measures — Sheldon Axler

Mathematics — Measure Theory · inverse-image-examples-sine

Problem

Verify the inverse image computations in Example 2.32: if

f : [0, 4\pi] \to \mathbf{R}

is defined by

f(x) = \sin x

, then

f^{-1}\bigl((0,\infty)\bigr) = (0,\pi) \cup (2\pi, 3\pi), \quad f^{-1}([0,1]) = [0,\pi] \cup [2\pi,3\pi] \cup \{4\pi\},

f^{-1}(\{-1\}) = \left\{\tfrac{3\pi}{2}, \tfrac{7\pi}{2}\right\}, \quad f^{-1}\bigl((2,3)\bigr) = \varnothing.