For f ∈ L¹_loc(ℝⁿ), what does the Lebesgue differentiation theorem assert?
AThe ball averages of f converge to f(x) for every x
BThe ball averages of f converge to f(x) for almost every x
Cf is differentiable almost everywhere
Df is continuous almost everywhere
The theorem concerns averages of f, not derivatives of f. The convergence holds outside a set of measure zero. An integrable function such as the indicator of the rationals is nowhere continuous, but its averages are all 0, matching the function almost everywhere.
Question 2 True / False
The Cantor function F is continuous and increasing on [0, 1] with F(0) = 0 and F(1) = 1. Since F'(x) = 0 almost everywhere, ∫ from 0 to 1 of F' equals F(1) - F(0).
TTrue
FFalse
Answer: False
∫ from 0 to 1 of F' = 0, but F(1) - F(0) = 1. The Cantor function is not absolutely continuous, so the Lebesgue version of the fundamental theorem of calculus does not apply to it.
Question 3 Short Answer
State the weak-type (1, 1) inequality for the Hardy-Littlewood maximal function on ℝⁿ.
Think about your answer, then reveal below.
Model answer: There is a constant C depending only on n (C = 3ⁿ works) such that |{x : Mf(x) > α}| ≤ (C/α)‖f‖₁ for every f ∈ L¹(ℝⁿ) and every α > 0.
The constant 3ⁿ comes from the Vitali covering lemma, which selects disjoint balls whose threefold enlargements cover the set. The estimate controls how large the set where averages are big can be, which is what the differentiation proof needs.
Question 4 Short Answer
If f ∈ L¹[a, b] and F(x) = ∫ from a to x of f(t) dt, what can be said about F'?
Think about your answer, then reveal below.
Model answer: F is differentiable almost everywhere on [a, b] and F'(x) = f(x) for almost every x.
The difference quotient (F(x + h) - F(x))/h is the average of f over an interval of length |h| next to x, so this is the one-dimensional case of the differentiation theorem, applied to one-sided intervals.