Exercise · ZW-MEASURABLE-FUNCTION-007 · ★★☆ Intermediate

Let \(f_n(x)=x^n\) on \([0,1]\). What is the pointwise limit \(f\) and why is it measurable?

Choose one option.

Show answer

B. \(f=\chi_{\{1\}}\), because it is a pointwise limit of measurable functions

Explanation

For x<1 the limit is 0 and at x=1 it is 1, so the limit is the indicator of {1}. It is measurable because pointwise limits of measurable functions are measurable.

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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-003 Which of these functions from ℝ to ℝ is NOT measurable with respect to the Borel σ-algebra?
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-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,∞))?