Measure, Integration & Real Analysis, 2021a

102 Chapter 4 Differentiation
4A
Hardy–Littlewood Maximal Function
Markov’s Inequality
The following result, called Markov’s inequality, has a sweet, short proof. We will
make good use of this result later in this chapter (see the proof of 4.10). Markov’s
inequality also leads to Chebyshev’s inequality (see Exercise 2 in this section).
4.1 Markov’s inequality
Suppose (X, S, μ) is a measure space and h ∈L 1 (μ). Then
for every c > 0.
μ({x ∈ X : |h(x)| ≥c}) ≤ 1 c ‖h‖ 1
Proof
Suppose c > 0. Then
μ({x ∈ X : |h(x)| ≥c}) = 1 c
≤ 1 c
∫
c dμ
{x∈X : |h(x)|≥c}
∫
|h| dμ
{x∈X : |h(x)|≥c}
≤ 1 c ‖h‖ 1,
as desired.
St. Petersburg University along the Neva River in St. Petersburg, Russia.
Andrei Markov (1856–1922) was a student and then a faculty member here.
CC-BY-SA A. Savin
Measure, Integration & Real Analysis, by Sheldon Axler