Orthomodular lattices, natural density and
non-distributive Lp spaces
Relations Between Banach Space Theory
and Geometric Measure Theory
Jarno Talponen
University of Eastern Finland
talponen@iki.fi
June, 2015
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
1 / 12
Abstract
The undercurrent in the paper involves orthomodular lattices and
generalized measure algebras where one replaces Boolean algebra & a
measure with a lattice & a submeasure.
In the first part of the talk we take a look at natural density of natural
numbers and how it can be related to measure algebras.
The second part of the paper and talk are speculative in nature. We
discuss how Lp spaces on lattices with submeasures ‘should’ look like.
Then the ‘supports’ of simple functions do not behave distributively as in
the Boolean case.
The preprint is available at ArXiv.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
2 / 12
Lattice
Lattice L = (L, ∨, ∧, 0, 1) is axiomatized quite similarly as a Boolean
algebra. Typically Boolean algebras and lattices differ in
complementation and distributivity; the latter does not need to enjoy
such properties.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
3 / 12
Lattice
Lattice L = (L, ∨, ∧, 0, 1) is axiomatized quite similarly as a Boolean
algebra. Typically Boolean algebras and lattices differ in
complementation and distributivity; the latter does not need to enjoy
such properties.
However, a lattice may have weaker versions of complementation
operation A 7→ A⊥ and that of distributivity. An example of such a
structure: orthomodular lattice.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
3 / 12
Lattice
Lattice L = (L, ∨, ∧, 0, 1) is axiomatized quite similarly as a Boolean
algebra. Typically Boolean algebras and lattices differ in
complementation and distributivity; the latter does not need to enjoy
such properties.
However, a lattice may have weaker versions of complementation
operation A 7→ A⊥ and that of distributivity. An example of such a
structure: orthomodular lattice.
A model for such a lattice: the system of closed subspaces of a
Hilbert space.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
3 / 12
Lattice
Lattice L = (L, ∨, ∧, 0, 1) is axiomatized quite similarly as a Boolean
algebra. Typically Boolean algebras and lattices differ in
complementation and distributivity; the latter does not need to enjoy
such properties.
However, a lattice may have weaker versions of complementation
operation A 7→ A⊥ and that of distributivity. An example of such a
structure: orthomodular lattice.
A model for such a lattice: the system of closed subspaces of a
Hilbert space.
A generalization of measure: An order-preserving map ϕ : L → [0, 1]
with ϕ(0) = 0, ϕ(1) = 1,
ϕ(A ∨ B) ≤ ϕ(A) + ϕ(B),
A, B ∈ L
is called a submeasure.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
3 / 12
Lattice
Lattice L = (L, ∨, ∧, 0, 1) is axiomatized quite similarly as a Boolean
algebra. Typically Boolean algebras and lattices differ in
complementation and distributivity; the latter does not need to enjoy
such properties.
However, a lattice may have weaker versions of complementation
operation A 7→ A⊥ and that of distributivity. An example of such a
structure: orthomodular lattice.
A model for such a lattice: the system of closed subspaces of a
Hilbert space.
A generalization of measure: An order-preserving map ϕ : L → [0, 1]
with ϕ(0) = 0, ϕ(1) = 1,
ϕ(A ∨ B) ≤ ϕ(A) + ϕ(B),
A, B ∈ L
is called a submeasure.
Relevance: Quantum theory.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
3 / 12
Natural density sets
Let us denote by D the collection of all density sets, i.e. sets A ⊂ N
such that the natural density
|A ∩ {1, . . . , n}|
n→∞
n
d(A) := lim
exists. We denote by N ⊂ D the collection of all null density sets, i.e.
sets A with d(A) = 0. Equivalence relation: A ∼ B if A M B ∈ N .
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
4 / 12
Natural density sets
Let us denote by D the collection of all density sets, i.e. sets A ⊂ N
such that the natural density
|A ∩ {1, . . . , n}|
n→∞
n
d(A) := lim
exists. We denote by N ⊂ D the collection of all null density sets, i.e.
sets A with d(A) = 0. Equivalence relation: A ∼ B if A M B ∈ N .
Important in number theory. Have been studied in Banach spaces too.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
4 / 12
Natural density sets
Let us denote by D the collection of all density sets, i.e. sets A ⊂ N
such that the natural density
|A ∩ {1, . . . , n}|
n→∞
n
d(A) := lim
exists. We denote by N ⊂ D the collection of all null density sets, i.e.
sets A with d(A) = 0. Equivalence relation: A ∼ B if A M B ∈ N .
Important in number theory. Have been studied in Banach spaces too.
Notoriously badly behaved: A ∩ B, A ∪ B may fail to be density sets
even if A and B are such.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
4 / 12
Natural density sets
Let us denote by D the collection of all density sets, i.e. sets A ⊂ N
such that the natural density
|A ∩ {1, . . . , n}|
n→∞
n
d(A) := lim
exists. We denote by N ⊂ D the collection of all null density sets, i.e.
sets A with d(A) = 0. Equivalence relation: A ∼ B if A M B ∈ N .
Important in number theory. Have been studied in Banach spaces too.
Notoriously badly behaved: A ∩ B, A ∪ B may fail to be density sets
even if A and B are such.
Singletons have density 0, thus σ-additivity of d fails.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
4 / 12
A result on turning a system of density sets to a measure
algebra
But if the previous obstructions are dealt with, then the measure sets
turn to measure algebras.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
5 / 12
A result on turning a system of density sets to a measure
algebra
But if the previous obstructions are dealt with, then the measure sets
turn to measure algebras.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
5 / 12
A result on turning a system of density sets to a measure
algebra
But if the previous obstructions are dealt with, then the measure sets
turn to measure algebras.
Theorem
Let F ⊂ D be a family closed under finite intersections. Then there is a
σ-algebra Σ order-isomorphically included in D/∼ such that F/∼ ⊂ Σ and
d̂ : Σ → [0, 1], d̂(K /∼ ) = d(K ), is σ-additive.
Morerover, if F/∼ is countable and the corresponding σ-generated
measure algebra (Σ, d̂) is atomless, then it is in fact isomorphic to the
measure algebra on the unit interval.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
5 / 12
Structure of the proof 1/2: σ-additivity of d on the cosets
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
6 / 12
Structure of the proof 1/2: σ-additivity of d on the cosets
Lemma
With the above notations the mapping d̂ is countably additive in the
following sense: Suppose
that ([Ak ])k∈N ⊂ D/
-increasing
W
W∼ is a N W
sequence. Then k [Ak ] ∈ D/∼ exists and d̂( k [Ak ]) = k d̂([Ak ]).
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
6 / 12
Structure of the proof 1/2: σ-additivity of d on the cosets
Lemma
With the above notations the mapping d̂ is countably additive in the
following sense: Suppose
that ([Ak ])k∈N ⊂ D/
-increasing
W
W∼ is a N W
sequence. Then k [Ak ] ∈ D/∼ exists and d̂( k [Ak ]) = k d̂([Ak ]).
The gist of the proof: ‘lagged’ subsets.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
6 / 12
Structure of the proof 1/2: σ-additivity of d on the cosets
Lemma
With the above notations the mapping d̂ is countably additive in the
following sense: Suppose
that ([Ak ])k∈N ⊂ D/
-increasing
W
W∼ is a N W
sequence. Then k [Ak ] ∈ D/∼ exists and d̂( k [Ak ]) = k d̂([Ak ]).
The gist of the proof: ‘lagged’ subsets.
Version of the main lemma in the paper in more general form:
N
L, [0, 1]
G , {1, . . . , n}-averages
ϕn , lim
limF .
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
6 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i) [N] ∈ ∆;
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i) [N] ∈ ∆;
(ii) For each [A], [B] ∈ ∆, [A] N [B], we have [B \ A] ∈ ∆;
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i) [N] ∈ ∆;
(ii) For each [A], [B] ∈ ∆, [A] N [B], we have [B \ A] ∈ ∆;
(iii) For each [A], [B] ∈ ∆, A ∩ B = ∅, we have [A ∪ B] ∈ ∆;
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i)
(ii)
(iii)
(iv)
[N] ∈ ∆;
For each [A], [B] ∈ ∆, [A] N [B], we have [B \ A] ∈ ∆;
For each [A], [B] ∈ ∆, A ∩ B = ∅, we have [A ∪ B] ∈ ∆;
If (Ak ) ⊂ ∆ is an increasing sequence in the order inherited from D/∼ ,
then the least upper bound A for this sequence exists in D/∼ and
moreover A ∈ ∆.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i)
(ii)
(iii)
(iv)
[N] ∈ ∆;
For each [A], [B] ∈ ∆, [A] N [B], we have [B \ A] ∈ ∆;
For each [A], [B] ∈ ∆, A ∩ B = ∅, we have [A ∪ B] ∈ ∆;
If (Ak ) ⊂ ∆ is an increasing sequence in the order inherited from D/∼ ,
then the least upper bound A for this sequence exists in D/∼ and
moreover A ∈ ∆.
Let ∆ be the intersection of all d-systems in D/∼ containing F/∼ .
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Structure of the proof 2/2: Dynkin lemma argument
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We
call a subset ∆ ⊂ D/∼ a d-system if it satisfies the following
conditions:
(i)
(ii)
(iii)
(iv)
[N] ∈ ∆;
For each [A], [B] ∈ ∆, [A] N [B], we have [B \ A] ∈ ∆;
For each [A], [B] ∈ ∆, A ∩ B = ∅, we have [A ∪ B] ∈ ∆;
If (Ak ) ⊂ ∆ is an increasing sequence in the order inherited from D/∼ ,
then the least upper bound A for this sequence exists in D/∼ and
moreover A ∈ ∆.
Let ∆ be the intersection of all d-systems in D/∼ containing F/∼ .
The modification of the π-λ-lemma argument gives that ∆ is
essentially a σ-algebra.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
7 / 12
Spaces of the type Lp (L, ϕ)
Speculative part of the paper. What is Lp (L, ϕ) where ϕ is a
submeasure on a lattice L?
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
8 / 12
Spaces of the type Lp (L, ϕ)
Speculative part of the paper. What is Lp (L, ϕ) where ϕ is a
submeasure on a lattice L?
It should be the completion of a suitable normed space of ‘simple
functions’. Except that the space is pointless - not a genuine function
space.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
8 / 12
Spaces of the type Lp (L, ϕ)
Speculative part of the paper. What is Lp (L, ϕ) where ϕ is a
submeasure on a lattice L?
It should be the completion of a suitable normed space of ‘simple
functions’. Except that the space is pointless - not a genuine function
space.
That is mimicking a possible (re)construction of Lp (µ) from the
measure algebra (A, µ), where A = Σ/Ker(µ).
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
8 / 12
Spaces of the type Lp (L, ϕ)
Speculative part of the paper. What is Lp (L, ϕ) where ϕ is a
submeasure on a lattice L?
It should be the completion of a suitable normed space of ‘simple
functions’. Except that the space is pointless - not a genuine function
space.
That is mimicking a possible (re)construction of Lp (µ) from the
measure algebra (A, µ), where A = Σ/Ker(µ).
We start with the space c00 (L) and denote its canonical Hamel basis
unit vectors by eA , A ∈ L.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
8 / 12
Spaces of the type Lp (L, ϕ)
Speculative part of the paper. What is Lp (L, ϕ) where ϕ is a
submeasure on a lattice L?
It should be the completion of a suitable normed space of ‘simple
functions’. Except that the space is pointless - not a genuine function
space.
That is mimicking a possible (re)construction of Lp (µ) from the
measure algebra (A, µ), where A = Σ/Ker(µ).
We start with the space c00 (L) and denote its canonical Hamel basis
unit vectors by eA , A ∈ L.
This vector space by itself is not ‘realistic’ model for ‘simple
functions’ because there is a spike supported on 0 (empty set). Also,
c00 (L) does not recognize the possible overlap of the supports.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
8 / 12
Tensorification
To fix these issues we can use an approach similar to the tensor space
constructions.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
9 / 12
Tensorification
To fix these issues we can use an approach similar to the tensor space
constructions.
Let ∆ ⊂ c00 (L) be the linear subspace given by
∆ = [e0 ] + span((eA + eB ) − (eA∨B + eA∧B ) : A, B ∈ L}.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
9 / 12
Tensorification
To fix these issues we can use an approach similar to the tensor space
constructions.
Let ∆ ⊂ c00 (L) be the linear subspace given by
∆ = [e0 ] + span((eA + eB ) − (eA∨B + eA∧B ) : A, B ∈ L}.
We let
q : c00 (L) → c00 (L)/∆
be the canonical quotient mapping.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
9 / 12
Tensorification
To fix these issues we can use an approach similar to the tensor space
constructions.
Let ∆ ⊂ c00 (L) be the linear subspace given by
∆ = [e0 ] + span((eA + eB ) − (eA∨B + eA∧B ) : A, B ∈ L}.
We let
q : c00 (L) → c00 (L)/∆
be the canonical quotient mapping.
Write X = c00 (L)/∆ and we denote by
a ⊗ A := q(aeA ) ∈ X ,
a ∈ R, A ∈ L.
Note that X is the space of vectors of the form
X
ai ⊗ Ai , ai ∈ R, Ai ∈ L, I finite.
i∈I
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
9 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
10 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
(i) v satisfies the axioms of a partial order of a vector lattice, except
possibly anti-symmetry;
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
10 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
(i) v satisfies the axioms of a partial order of a vector lattice, except
possibly anti-symmetry;
(ii) a ⊗ A v b ⊗ A whenever a ≤ b;
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
10 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
(i) v satisfies the axioms of a partial order of a vector lattice, except
possibly anti-symmetry;
(ii) a ⊗ A v b ⊗ A whenever a ≤ b;
(iii) 1 ⊗ A v 1 ⊗ B whenever A ≤ B in the intrinsic partial order of L.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
10 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
(i) v satisfies the axioms of a partial order of a vector lattice, except
possibly anti-symmetry;
(ii) a ⊗ A v b ⊗ A whenever a ≤ b;
(iii) 1 ⊗ A v 1 ⊗ B whenever A ≤ B in the intrinsic partial order of L.
We define a semi-norm on X by
!
ρ
X
ai ⊗ Ai
i
= inf

 X

Jarno Talponen (UEF)
k
!1
p
|bk |p ϕ(Bk )
: ±
X
ai ⊗ Ai v
i
On generalized measure algebras
X
|bk | ⊗ Bk


.

k
June, 2015
10 / 12
Auxiliary vector preorder
Let v be the preorder on X generated by the following conditions:
(i) v satisfies the axioms of a partial order of a vector lattice, except
possibly anti-symmetry;
(ii) a ⊗ A v b ⊗ A whenever a ≤ b;
(iii) 1 ⊗ A v 1 ⊗ B whenever A ≤ B in the intrinsic partial order of L.
We define a semi-norm on X by
!
ρ
X
ai ⊗ Ai
i
= inf

 X

k
!1
p
|bk |p ϕ(Bk )
: ±
X
ai ⊗ Ai v
i
X
|bk | ⊗ Bk


.

k
Indeed, it is easy to see that this is a semi-norm; the triangle
inequality follows from the condition that x + y v v + w whenever
x v v and y v w .
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
10 / 12
Non-distributive Lp space
We define Lp (L, ϕ) to be the completion of (X /ker(ρ), ρ(·)).
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
11 / 12
Non-distributive Lp space
We define Lp (L, ϕ) to be the completion of (X /ker(ρ), ρ(·)).
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
11 / 12
Non-distributive Lp space
We define Lp (L, ϕ) to be the completion of (X /ker(ρ), ρ(·)).
Theorem
Let 1 ≤ p < ∞, L be an orthomodular lattice with an order-preserving
map ϕ : L → [0, 1], as above. Let (Ω, Σ, µ) be a probability space and
Σ0 ⊂ Σ a Boolean algebra which σ-generates Σ. Let us assume that
ϕ(M ∨ N) = ϕ(M) + ϕ(N) whenever N ≤ M ⊥ . Suppose that there is an
order-embedding  : Σ0 → L such that µ(M) = ϕ(M) for all M ∈ Σ0 .
(We are not assuming here that  respects the orthocomplementation
operation.) Then
X
X
ai [1Ai ]a.e.
→
7
ai ⊗ (Ai ), Ai ∈ Σ0
=
i
i
extends to a linear (into) isometry Lp (Ω, Σ, µ) → Lp (L, ϕ).
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
11 / 12
Some related references
A. Blass, R. Frankiewicz, G. Plebanek, C. Ryll-Nardzewski, A note on
extensions of asymptotic density. Proc. Amer. Math. Soc. 129 (2001),
3313–3320.
C.R. Buck, The measure theoretic approach to density. Amer. J. Math. 68,
(1946), 560–580.
E.F. Buck, C.R. Buck, A note on finitely-additive measures. Amer. J. Math.
69, (1947), 413–420.
D.H. Fremlin, Measure Theory. Torres Fremlin 2000.
P. de Lucia and P. Morales, A non-commutative version of a theorem of
Marczewski for submeasures. Studia Math, 101 (1992), 123–138.
M. Di Nasso, Fine asymptotic densities for sets of natural numbers. Proc.
Amer. Math. Soc. 138 (2010), 2657–2665.
Jarno Talponen (UEF)
On generalized measure algebras
June, 2015
12 / 12