Bounded Finitely Additive Measures

• Bounded finitely additive measures are set functions defined on algebras that are finitely additive and uniformly bounded by their total variation norm, generalizing classical measures.
• They underpin extensions of classical limit theorems and are integral to ergodic theory, martingale problems, and asset pricing models where countable additivity is relaxed.
• Their robust algebraic structure enables unique extensions to Boolean completions and facilitates representation in Banach lattices, impacting operator theory and functional analysis.

A bounded finitely additive measure is a set function defined on an algebra (or lattice) of subsets of a fixed set, which is additive over finite disjoint unions and possesses a uniform bound on its total variation or norm. Such measures generalize countably additive measures by relaxing the requirement of countable additivity, but they retain essential properties such as finite additivity and boundedness. They form a Banach lattice under the total variation norm and play a fundamental role in various areas including ergodic theory, martingale problems, operator theory, and abstract measure representation.

1. Formal Definition and Properties

Given a nonempty set $Q$ and an algebra $\mathscr A$ of its subsets, a real-valued set function $m\colon\mathscr A\to\mathbb{R}$ is finitely additive if:

• $m(\varnothing)=0$;
• $m(A\cup B)=m(A)+m(B)$ for $A,B\in\mathscr A$, $A\cap B=\varnothing$.

A measure $m\in \mathrm{ba}(\mathscr A)$ is bounded in variation if its norm

$$\|m\| := |m|(Q), \quad |m|(A) = \sup \Bigl\{ \sum_{i=1}^n |m(A_i)| : A_1,\dots,A_n \text{ partition } A \Bigr\}$$

is finite. The space $\mathrm{ba}(\mathscr{A})$ of all bounded finitely additive measures on $\mathscr{A}$ forms a Banach lattice whose norm is order-continuous, but not generally weakly compact or reflexive (Cassese, 2015). In the context of measure and integration over lattices, a similar inclusion-exclusion definition and boundedness constraint apply; for bounded lattices, additivity is defined according to the lattice operations (see (Massri et al., 2017)).

Boundedness of a finitely additive measure is essential: it ensures well-defined integration for bounded measurable functions, admits a dual Banach space structure, and is required for major theorems, such as extensions of the Komlós subsequence principle (Cassese, 2015).

2. The Role of Bounded Finitely Additive Measures in Limit Theorems

The structural study of $\mathrm{ba}(\mathscr A)$ and its subsequences underpins various generalizations of classical limit theorems. A Komlós-type subsequence theorem holds in $\mathrm{ba}(\mathscr{A})$: for any bounded sequence $(m_n)_{n\in\mathbb{N}}$ in $\mathrm{ba}(\mathscr{A})$, there exists a subsequence whose Cesàro means converge in the variation norm to an element $m\in\mathrm{ba}(\mathscr{A})$,

$$\lim_{K\to\infty}\left\| \frac{1}{K} \sum_{k=1}^K m_{n_k} - m \right\| = 0.$$

This extends the Cesàro-mean convergence principle of Komlós to the finitely additive context, revealing "hidden compactness" even though $\mathrm{ba}(\mathscr{A})$ is not weakly compact in general (Cassese, 2015). Norm boundedness is necessary for the result: if the sequence is not uniformly bounded, subsequential convergence need not hold.

Applications include:

• Strong laws of large numbers for sequences of integrals on finitely additive probability spaces;
• Merging of opinions and ergodic theorems in settings lacking regular conditional probabilities;
• Compactness and sequential structure theorems for $\mathrm{ba}(\mathscr{A})$ (Cassese, 2015).

3. Representation and Algebraic Structure

Bounded finitely additive measures can be characterized through algebraic and geometric tools, especially in the context of measures on bounded lattices. For a bounded lattice $X$, a bounded measure $\mu\colon X\to B$ (with $B$ an $A$-module) must satisfy normalization and a generalized inclusion-exclusion relation. The module of bounded measures $\mathcal{M}(X,B)$ is an additive functor, representable by

$$R_X=\mathbb{Z}[X] / \langle 0_X, 1_X-1, x \wedge y-xy, x \vee y-x-y+xy\rangle,$$

where $\pi:X \to R_X$ is the universal measure, and

$$\mathcal{M}(X,B) \cong \operatorname{Hom}_{\mathbb{Z}}(R_X, B).$$

Every bounded measure on $X$ extends uniquely to a bounded measure on a suitable Boolean lattice $Y$ generated by the idempotents in $R_X$ (Massri et al., 2017). This approach unifies classical measure representation theorems with scheme-theoretic perspectives, including the identification of $\mathcal{M}(X,B)$ as global sections of a sheaf over $\operatorname{Spec}(A)$.

For finite bounded lattices, the number of independent bounded measures corresponds to the rank of the representing ring, and classical enumerative results (e.g., via Möbius inversion) are recovered as special cases (Massri et al., 2017).

4. Martingale Problems and Applications in Asset Pricing

The theory of bounded finitely additive measures is pivotal in contexts where martingale measures fail to be countably additive. For a probability space $(\Omega,\mathcal{A},P_0)$ and the space $L$ of bounded $\mathcal{A}$-measurable functions, there exists a bounded finitely additive probability $P$ (possibly not countably additive) such that

$$P \sim P_0, \quad E_P[X]=0 \quad \forall\, X \in L,$$

if and only if there is a constant $c > 0$ and $Q\sim P_0$ countably additive with $c\,E_Q[X] \le \text{ess sup}(-X)$ for all $X\in L$ (Berti et al., 2010). The difference between equivalence and absolute continuity of $P$ is captured by simple conditions involving the essential supremum of $X$.

In mathematical finance, the existence of such $P$ provides a finitely additive analogue of the Fundamental Theorem of Asset Pricing. No-arbitrage is equivalent to the existence of an absolutely continuous finitely additive martingale measure, even if countably additive martingale measures fail to exist. The boundedness of $L$ is essential for the coherence arguments yielding these measures (Berti et al., 2010).

5. Ergodic Theory and Operator Applications

Bounded finitely additive measures are deeply intertwined with ergodic theorems and the spectral theory of Markov operators. In discrete-time homogeneous Markov chains, the Banach space $\mathrm{ba}(X,\Sigma)$ of bounded charges admits a dual Markov operator $A$,

$A\mu(E) = \int_X p(x,E)\,\mu(dx),$

and Cesàro averages over iterates display uniform ergodic behavior if and only if all invariant finitely additive measures are countably additive, i.e., absence of pure charges (Zhdanok, 2020). Under classical ergodicity conditions such as Doeblin’s condition, this is equivalent to the finite-dimensionality of the convex set of invariant measures.

In operator-theoretic frameworks, weak ergodic theorems are extended to Markov chains lacking invariant countably additive measures. The weak convergence of Cesàro means in the topology $\sigma(\mathrm{ba}(X),C(X))$ is characterized by inseparability of all invariant finitely additive measures from the limit measure. Regularization processes embed purely finitely additive phenomena into the structure of the Banach space $ba(X)$; examples illustrate non-invariance and non-countable additivity of limit measures (Zhdanok, 2018).

6. Integration, Measurability, and Functional Analysis

The integration theory for bounded finitely additive measures (charges) retains classical structures with significant modifications. For a bounded charge space $(X,\mathcal{A},\mu)$, the space $L_1(X,\mathcal{A},\mu)$ of integrable $T_1$-measurable functions is defined via limits of simple functions under "hazy" convergence, and norm completeness is characterized by the countable additivity of the completion of the measure on the quotient Boolean algebra $\mathcal{A}/\mathcal{N}$ (Keith, 2021).

For such bounded charge spaces:

• $T_1$-measurability is equivalent to $T_2$ and smoothness;
• Integrability is characterized by control over the tails of $|f|$ and the existence of a determining sequence of simple functions;
• $L_1$ need not be complete without additional algebraic properties of the algebra $\mathcal{A}$ or the measure.

When extended by Peano–Jordan completion, the equivalence classes of $L_p$ spaces remain unchanged, under mild assumptions (Keith, 2021).

7. Extension Theorems and Algebraic Extensions

Bounded finitely additive measures possess robust extension properties across algebraic and operator contexts. For example, every bounded finitely additive $X$-valued measure on the lattice of projections in a JBW$^*$-algebra admits a unique extension to a bounded linear operator on the algebra, provided there is no type $I_2$ summand; in the commutative case, this generalizes the Yosida–Hewitt decomposition (Escolano et al., 3 Sep 2025).

Similar extension and representation principles operate in bounded lattices, where bounded finitely additive measures extend uniquely to their Boolean completions and admit diverse scheme-theoretic interpretations (Massri et al., 2017).

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• "A Version of Komlós Theorem for Additive Set Functions" (Cassese, 2015)
• "On the representation of measures over bounded lattices" (Massri et al., 2017)
• "Finitely additive equivalent martingale measures" (Berti et al., 2010)
• "Weak ergodic theorem for Markov chains without invariant countably additive measures" (Zhdanok, 2018)
• "The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra" (Escolano et al., 3 Sep 2025)
• "A General Theorem of Gauß Using Pure Measures" (Schönherr et al., 2017)
• "Properties of functions on a bounded charge space" (Keith, 2021)
• "Ergodicity conditions for general Markov chains in terms of invariant finitely additive measures" (Zhdanok, 2020)
• "Finitely additive functions in measure theory and applications" (Alpay et al., 2022)