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 = 1j, j ∈ N∗, we apply Egorov’s theorem on each Ek, and find Ak,j ∈ τ such that µ (Ak,j) < 1j 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 (Ak,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 = 1j, ∀j ∈ N∗, which gives Aj such that µ(Aj) < 1j
and limn→+∞kfn− f kL∞(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 Nεα such that
n ∈ N and n ≥ Nεα ⇒ sup x∈X\Aε |fn(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, ∃Nεα∈ N : n ≥ Nεα⇒ 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, ∃Nεα∈ N : n ≥ Nεα ⇒ 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
⊂
x ∈ X : |fn(x) − f (x)| ≥ α 2 [ x ∈ X : |f (x) − fm(x)| ≥ α 2 . We can choose ε = ε 2i and α = 12i, ∀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\Ai: 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).
