Adriaan C. Zaanen

Linear Functionals on Spaces of Measurable Functions

As in the Examples 9.5 and 12.5(iii) we assume that μ is a σ-finite (non-negative and σ-additive) measure in the non-empty point set X and L0 = L0 (X, μ) is the Dedekind complete Riesz space of all μ-measurable (real) functions on X. To be precise, μ is defined on a σ-algebra Γ the members of which are called the μ-measurable subsets of X and the real function f on X is called μ-measurable whenever the set (x ∈ X: f (x) > α) is μ-measurable for every real α. It follows then easily that the sets of points x where f (x) ≥ α, f (x) < α, f (x) ≤ α or β ≤ f (x) ≤ α (for α and β real) respectively are also μ-measurable. Recall that functions differing only on a set of measure zero are identified, so that the members of L0 are in fact equivalence classes of measurable functions. Similarly for the members of Γ (as explained already in Example 3.4(ii)).

Order Continuous Operators

We shall now turn our attention to a special kind of linear operators mapping a Riesz space E into another Riesz space F,called order continuous operators. As long as nothing more is assumed about F the interest is focused mainly on positive operators, but if F is Dedekind complete any regular operator T : E → F has an absolute value | T |, and then we can say more. We begin by presenting the definition of an order continuous operator.

Results with the Hahn-Banach Theorem

This chapter is devoted to several results the proofs of which are based on a theorem known as the Hahn-Banach theorem (due to H. Hahn, 1927, and S. Banach, 1929). Most of these results, as well as the Hahn-Banach theorem itself, are extension theorems dealing with extension of a (linear) operator from a linear subspace to the entire space, thereby preserving certain properties of the operator (such as, for example, positivity or norm boundedness). To prove the Hahn-Banach theorem it is necessary to accept a certain axiom about partially ordered sets, called Zorn’s lemma. We formulate the axiom. The definition of a chain and of a maximal element in a partially ordered set are given in section 1.

Archimedean Spaces and Convergence

The Riesz space E is said to be Archimedean if inf (n-1u: n = 1, 2,…) = 0 holds for every u ∈ E + .There exist non-Archimedean Riesz spaces. As an example, let E= ℝ2 with the lexicographical ordering (see Example 4.2(ii)). The element (0,1) in E is a lower bound of the sequence (n-1, n-1): n = 1, 2,…. Hence, u = (1,1) does not satisfy the condition that inf(n-lu: n = 1, 2,…) = 0. Actually, the sequence of all n-1u does not have an infimum at all in this case. Before presenting more examples, we prove a simple theorem.

Signed Measures and the Radon-Nikodym Theorem

Recall that a non-empty collection Γ of subsets of the set X is called a ring of subsets of X if A ∈ Γ, B ∈ Γ implies A ∪ B ∈ Γ and A\B ∈ Γ (see Example 3.4(i)). It follows then that finite unions and finite intersections of sets in Γ are sets in Γ. The ring Γ is called an algebra of subsets of X if X itself is a member of Γ. The algebra Γ is called a σ-algebra if any countable union of sets in Γ is again a set in Γ (see Example 3.4(ii)). The mapping v from the algebra Γ into ℝ is called a (real) finitely additive signed measure (or a charge) on Γ if v(A1 ∪ A2) = v (A1) + v (A2) holds for all disjoint Al and A2 in Γ (see Example 4.3(7)). If Γ is a σ-algebra and v \( \left( { \cup _1^\infty {A_n}} \right) = \sum\nolimits_1^\infty {v\left( {{A_n}} \right)} \)holds for every disjoint sequence (A n : n = 1, 2,…) in Γ, then v is called a σ-additive signed measure on Γ. In this section we shall briefly say “signed measure” when a σ-additive signed measure is meant. If there is only finite additivity, this will be explicitly mentioned.

Freudenthal’s Spectral Theorem

Let μ be a σ-finite measure in the (non-empty) point set X. As well-known, any finite linear combination of characteristic functions of measurable subsets of X is called a μ-step function. It is not difficult to see that for any 0 ≤ f ∈ L ∞ (X, μ) there exists a sequence (s n : n = 1, 2,…) of μ-step functions such that 0 ≤ s n ↑ f holds uniformly. A similar result holds in an Archimedean Riesz space possessing a strong unit. This result, due to H.Freudenthal (1936), is known as Freudenthal’s spectral theorem. Some preliminary theorems about projection bands will be useful. Recall that the band B in the Archimedean Riesz space E is called a projection band if E = B ⊕B d (see section 11). In this case B d is likewise a projection band (we have B = B dd because E is Archimedean, so E = B d ⊕ B dd ). As in Theorem 11.4, the band projection on the projection band B is sometimes denoted by P B . It is obvious that\( {P_{{B^d}}} = I - {P_B} \),where I is the identity operator in E (i.e., I f = f for each f ∈ E, and so\( {P_{{B^d}}}f = f - {P_B}f \)). The following holds now.

Normed Riesz Spaces and Banach Lattices

Let V be a real or complex vector space and assume that to each element f ∈ V there is assigned a real number ‖ f ‖ such that

Ideals, Bands and Disjointness

Let E be a Riesz space. Certain linear subspaces of E have special properties and have received special names. Details are contained in the following definition.

Carriers of Operators

In this brief section we discuss the notion of order denseness. This could have happened earlier, but there was no need for that.

Order Bounded Operators

In the present section we shall prove an important result which lies at the foundations of operator theory in Riesz spaces. Briefly stated, it says that if E and F are Riesz spaces with F Dedekind complete and T : E → F is a linear operator, then T is regular if and only if T is order bounded. In other words, using the notations introduced in section 18, we have L r (E, F) = L b (E, F). Moreover (and this is the really interesting point), the vector space L b (E, F) is now a Dedekind complete Riesz space with the set of positive operators (from E into F) as positive cone. This implies that every T ∈ L b (E,F) can be written as T = T + - T - , where T + and T - are positive operators such that T + =T∨0, T - = (-T) ∨0 and T+∧T-= 0. The case that F = ℝ is of special interest. Since ℝ is Dedekind complete, it follows that regular linear functionals on E are the same as order bounded linear functionals and the space L r (E, ℝ) = L b (E, ℝ) is a Dedekind complete Riesz space. This space, denoted by E~ for convenience, is called the order dual of E. The theorem stating that L r (E, F) = L b (E, F) is a Dedekind complete Riesz space is due to L.V. Kantorovitch (1936) in the Soviet Union and to H. Freudenthal (1936) in the Netherlands. The theorem on E~, with further results on order continuous linear functionals (see the next section), is due to F. Riesz (1937) in Hungary who, already in 1928 at the Bologna International Mathematical Congress, presented his ideas about linear functionals on certain ordered vector spaces, which is one of the reasons why lattice ordered vector spaces are now known as Riesz spaces.