Exercise · ZW-MEASURABLE-FUNCTION-003 · ★★☆ Intermediate
Which of these functions from ℝ to ℝ is NOT measurable with respect to the Borel σ-algebra?
Choose one option.
Show answer
- B. The indicator of a set that is not Borel
Explanation
The continuous function, the Dirichlet function and \(x^2\) are measurable because their preimages of intervals are Borel. The indicator of a non-Borel set has preimage of \((1/2,\infty)\) equal to that set, which is not Borel.
Review the concept: Measurable function
A function f: X → ℝ is measurable with respect to a σ-algebra \(\mathcal{A}\) on X if the preimage \(f^{-1}(B)\) belongs to \(\mathcal{A}\) for every Borel set B of ℝ.
\[f^{-1}(B)\in\mathcal{A}\ \forall B\in\mathcal{B}(\mathbb{R})\]
More practice problems: measurable function
- ZW-MEASURABLE-FUNCTION-001 What is the name of the condition satisfied by f when f^-1(B) lies in A for every Borel B?
- ZW-MEASURABLE-FUNCTION-002 Which σ-algebra is usually used on ℝ to define measurability of real functions?
- ZW-MEASURABLE-FUNCTION-004 What kind of set is the preimage f^-1((a,∞)) when f is measurable?
- ZW-MEASURABLE-FUNCTION-005 Is every continuous function from ℝ to ℝ measurable with respect to the Borel σ-algebra?
- ZW-MEASURABLE-FUNCTION-006 Let f(x)=begincases1 & x≥ 0\\ 0 & x<0endcases on ℝ. What is f^-1((1/2,∞))?
- ZW-MEASURABLE-FUNCTION-007 Let f_n(x)=x^n on [0,1]. What is the pointwise limit f and why is it measurable?
- ZW-MEASURABLE-FUNCTION-008 If f and g are measurable, why is f+g measurable?
- ZW-MEASURABLE-FUNCTION-009 Let f be measurable with f^-1((a,∞))inA for all a∈ℝ. What does this imply about f^-1([a,∞))?