About Egoroff s theorem

About Egorov’s theorem

Sylvain Dotti

May 11, 2018

Abstract

Egorov’s theorem, Lusin’s extension, relations with convergence in measure and almost everywhere convergence.

1

About him

Dmitri Egorov was a russian mathematician of the beginning of the XXth century. One of his student was Nicolai Lusin. Egorov was worshipping the name of God and treated the name of God as God Himself. For this belief, in the time of communism, he was sent to jail and died from his hunger-strike. Hard life...

2

The theorem (taken from Andr´

e Gramain’s book [

Gra94

])

Let (X, τ, µ) be a finite positive mesure space.

If (fn)n∈N is a sequence of measurable functions fn : X → R, converging µ-almost

everywhere towards a measurable function f : X → R, then ∀ε > 0, ∃A ∈ τ : µ(A) < ε and lim

n→+∞kfn− f kL∞(X\A)= 0.

3

One proof

For k, n ∈ N∗, we can define the subsets of X : An,k = n x ∈ X : |fn(x) − f (x)| > 1 k o

At fixed k, the upper limit on n of An,k is τ -measurable and

µ  lim sup n→+∞ An,k  = µ   \ n∈N∗ [ i≥n Ai,k  = 0.

Since µ is non-negative and finite, since the sequence   [ i≥n Ai,k   n∈N∗ is non-increasing, we have lim n→+∞   [ i≥n Ai,k  = 0.

So, there is a Nk ∈ N∗ such that

µ   [ i≥Nk Ai,k  < ε 2k. The set A = [ k∈N∗ [ i≥Nk Ai,k

is of measure less than ε and for all

x ∈ X\A = \

k∈N∗

\

i≥Nk

X\Ai,k, we have ∀k, ∃Nk : ∀n ≥ Nk, |fn(x) − f (x)| ≤

1 k.

4

A counter-exemple if the positive measure is not bounded

Take X = N, τ = P (N) , µ(A) = card(A) ∀A ∈ τ, fn= 1{0,...,n}, f = 1N, then

|fn(x) − f (x)| = 1, ∀x ∈ N\{1, ..., n}.

5

An extension of the Egorov theorem due to Lusin (see

Saks [

Sak37

] p19)

Suppose (X, τ, µ) is a σ-finite measure space, i.e. there exists (Ek)_k∈N such that

µ (Ek) < +∞, ∀k ∈ N and

X = [

k∈N

Ek.

Let (fn)n∈N be a sequence of measurable functions fn : X → R, converging µ-almost

everywhere towards a measurable function f : X → R, that is ∃H ∈ τ and µ(H) = 0 such that

For ε = 1_j, j ∈ N∗, we apply Egorov’s theorem on each Ek, and find Ak,j ∈ τ such that µ (Ak,j) < 1_j and lim n→+∞kfn− f kL∞(Ek\Ak,j)= 0. Then Ek=   [ j∈N∗ (Ek\Ak,j)   [   \ j∈N∗ Ak,j  . As µ   \ j∈N∗ Ak,j  ≤ µ (A_k,N) ≤ 1 N, ∀N ∈ N ∗ , the set T j∈N∗Ak,j is of measure 0. Conclusion : X =   [ k∈N [ j∈N∗ (Ek\Ak,j)   [   [ k∈N \ j∈N∗ Ak,j   and lim n→+∞kfn− f kL∞(Ek\Ak,j)= 0, ∀k, j.

6

The Egorov property gives a caracterisation of almost

everywhere convergence

Let (X, τ, µ) be a positive mesure space.

Let (fn)n∈Nbe a sequence of measurable functions fn: X → R, f : X → R a measurable

function, such that

∀ε > 0, ∃A ∈ τ : µ(A) < ε and lim

n→+∞kfn− f kL∞(X\A)= 0,

then (fn)_n∈N converges µ-almost everywhere towards f .

Proof :

We can use the Egorov property for ε = 1_j, ∀j ∈ N∗, which gives Aj such that µ(Aj) < 1_j

and limn→+∞kfn− f k_L∞_(X\A

j)= 0. By writing X =   [ j∈N∗ X\Aj   [   \ j∈N∗ Aj  . we can conclude.

7

Definition of convergence in measure

Let (X, τ, µ) be a positive measure space.

We say that a sequence (fn)n∈N of mesurable functions fn : X → R, is converging in

measure towards the measurable function f : X → R if ∀ε > 0, lim

n→+∞µ x ∈ X : |fn(x) − f (x)| ≥ ε
= 0.

8

The Egorov property implies convergence in measure

Let (X, τ, µ) be a positive measure space.

Let (fn)n∈Nbe a sequence of measurable functions fn: X → R, f : X → R a measurable

function, such that

∀ε > 0, ∃A ∈ τ : µ(A) < ε and lim

n→+∞kfn− f kL∞(X\A)= 0,

then (fn)_n∈N converges in measure towards f .

Proof :

Let’s fix α, ε > 0. The assumption says that there exists a set Aε∈ τ such that µ (Aε) < ε

and an integer N_εα such that

n ∈ N and n ≥ Nεα ⇒ sup x∈X\Aε |f_n(x) − f (x)| < α It means that x ∈ X : |fn(x) − f (x)| ≥ α ⊂ Aε and µ x ∈ X : |fn(x) − f (x)| ≥ α
≤ µ (Aε) < ε. Finally, ∀ε > 0, ∃N_εα_{∈ N : n ≥ Nε}α⇒ µ x ∈ X : |fn(x) − f (x)| ≥ α
< ε.

9

Convergence in measure implies the Egorov property for

a subsequence

towards the measurable function f : X → R. Then, there exists a subsequence (fni)i∈N

such that

∀ε > 0, ∃A ∈ τ : µ(A) < ε and lim

i→+∞kfni− f kL∞(X\A) = 0.

Proof (taken from Hartman and Mikusinski’s book [HM61]): Let (fn)n∈N converge in measure towards f , then

∀ε, α > 0, ∃N_εα_{∈ N : n ≥ Nε}α ⇒ µ x ∈ X : |fn(x) − f (x)| ≥ α
< ε. It implies that ∀ε, α > 0, ∃Nα ε ∈ N : m, n ≥ Nεα ⇒ µ x ∈ X : |fn(x) − fm(x)| ≥ α
< ε because x ∈ X : |fn(x) − fm(x)| ≥ α

⊂

2i, ∀i ∈ N, and a subsequence (fni)i∈N such that

∀i ∈ N, n ≥ ni ⇒ µ  x ∈ X : |fn(x) − fni(x)| ≥ 1 2i  < ε 2i Naming Ai=x ∈ X :  fni+1(x) − fni(x)  ≥ 1 2i , ∀i ∈ N ∗_, we have ∀x ∈ X\A_i:fni+1(x) − fni(x)  < 1 2i and µ(Ai) < ε 2i.

The subsequence (fni)_i∈N is converging uniformly towards f on X\

[ i∈N Ai and we have µ [ i∈N Ai ! < 4ε. Indeed, if x ∈ X\[ i∈N Ai, ∀k ∈ N∗ we have  fni+k(x) − fni(x)  < 1 2i + 1 2i+1+ ... + 1 2i+k−1 < 1 2i−1.

The conclusion is obvious because L∞(X\S

10

Some remarks

• Instead of lim

n→+∞kfn− f kL∞(X\A) = 0, many authors prefer to write fn→n→+∞f

uniformly on X\A in the Egorov property

• What I call the Egorov property is often called the almost uniform convergence. • There are other Egorov’s extensions, see for example [Bar80] or [Rob16].

References

[Bar80] Robert G Bartle. “An extension of Egorov’s theorem”. In: The American Math-ematical Monthly 87.8 (1980), pp. 628–633 (cit. on p.6).

[Gra94] Andr´e Gramain. Integration. Hermann, 1994 (cit. on p. 1).

[HM61] Stanislaw Hartman and Jan Mikusinski. The Theory of Lebesgue Measure and Integration: Transl. from Polish by Leo F. Boron. PWN, 1961 (cit. on p.5). [Rob16] Mangatiana A Robdera. “Extensions of the Lusin’s Theorem, the

Severini-Egorov’s Theorem and the Riesz Subsequence Theorems”. In: (2016) (cit. on p.6).

[Sak37] Stanislas Saks. “Theory of the integral (Monografie Matematyczne 7)”. In: Vol. VII., Warszawa-Lwow (1937) (cit. on p.2).