26.5. RADEMACHER’S THEOREM 949

With the above lemmas, the following is the main theorem about absolutely continuousfunctions.

The following simple corollary is a case of Rademacher’s theorem.

Corollary 26.4.3 Suppose f : [a,b]→ R is Lipschitz continuous,

| f (x)− f (y)| ≤ K |x− y| .

Then f ′ (x) exists a.e. and

f (x) = f (a)+∫ x

af ′ (t)dt.

Proof: If f were increasing, this would follow from the above lemma. Let g(x) =2Kx− f (x) . Then g is Lipschitz with a different Lipschitz constant and also if x < y,

g(y)−g(x) = 2Ky− f (y)− (2Kx− f (x))

≥ 2K (y− x)−K |y− x|= k |y− x| ≥ 0

and so Lemma 26.4.2 applies to g and this shows that f ′ (t) exists for a.e. t and g′ (x) =2K− f ′ (x) . Also

2K (x−a)− ( f (x)− f (a))

= g(x)−g(a) = 2Kx− f (x)− (2Ka− f (a)) =∫ x

a

(2K− f ′ (t)

)= 2K (x−a)−

∫ x

af ′ (t)dt

showing that f (x)− f (a) =∫ x

a f ′ (t)dt.

26.5 Rademacher’s TheoremTo begin with is a useful proposition which says the the set where a sequence converges isa measurable set.

Proposition 26.5.1 Let { fn} be measurable functions with values in a complete normedvector space. Let A≡ {ω : { fn (ω)} converges} . Then A is measurable.

Proof: The set A is the same as the set on which { fn (ω)} is a Cauchy sequence. Thisset is

∩∞n=1∪∞

m=1∩p,q>m

[∥∥ fp (ω)− fq (ω)∥∥< 1

n

]which is a measurable set thanks to the measurability of each fn.

It turns out that Lipschitz functions on Rp can be differentiated a.e. This is calledRademacher’s theorem. It also can be shown to follow from the Lebesgue theory of differ-entiation. We denote Dv f (x) the directional derivative of f in the direction v. Here v is aunit vector. In the following lemma, notation is abused slightly. The symbol f (x+tv) willmean t→ f (x+tv) and d

dt f (x+tv) will refer to the derivative of this function of t.