Use the formal definition of measurability of a real-valued function to prove that thefollowing functions are measurable with respect to the Borel σ-algebra on R.(a) f (x) = x2 (b) g(x) = tan x (c) h(x) = −√x.
Question
Use the formal definition of measurability of a real-valued function to prove that thefollowing functions are measurable with respect to the Borel σ-algebra on R.(a) f (x) = x2 (b) g(x) = tan x (c) h(x) = −√x.
Solution
To prove that a function is measurable with respect to the Borel σ-algebra on R, we need to show that for every open set O in R, the preimage of O under the function is in the Borel σ-algebra. The Borel σ-algebra is the smallest σ-algebra containing all open intervals in R.
(a) f(x) = x^2
For any open set O in R, we need to show that f^(-1)(O) is a Borel set. Since f is continuous, the preimage of an open set under f is open. And every open set in R is a Borel set. Therefore, f is Borel-measurable.
(b) g(x) = tan x
The function tan x is continuous wherever it is defined. Therefore, the preimage of an open set under tan x is open, and hence a Borel set. However, tan x is not defined at odd multiples of π/2. But the set of points where tan x is not defined is a countable set, and every countable set is a Borel set. Therefore, g is Borel-measurable.
(c) h(x) = -√x
The function -√x is continuous for x ≥ 0. Therefore, the preimage of an open set under -√x is open in [0, ∞), and hence a Borel set. For x < 0, h(x) is not defined, but the set of such x is an open set, and hence a Borel set. Therefore, h is Borel-measurable.
Similar Questions
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ormal definition of measurability of a real-valued function
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