Contents of Measure Theory, by D.H.Fremlin

Chapter 64: The fundamental theorm of martingales and the S-integral

641 Previsible versions
The algebras Aτ-; the previsible version u< of a moderately oscillatory process; jumps and residual oscillations; the previsible version of an indefinite integral; quadratic variations; the previsible version of a previsible version; integrating a previsible version.

642 Previsible processes
Previsible processes; and previsible versions; previsible σ-algebras; previsibly measurable processes; and order*-convergence.

643 The fundamental theorem of martingales
uτ- and the conditional expectation Pτ-; the region of accessibility of a stopping time; previsible variations; assembling processes from components on stopping-time intervals; sublattices with countable cofinality are adequate; the fundamental theorem.

644 Pointwise convergence
A kind of sequential smoothness for the Riemann-sum integral; a weak topology on Mn-s; sufficient conditions to ensure that order*-convergence of ⟨un<n∈N implies convergence of ⟨∫undvn∈N.

645 Construction of the S-integral
Previsibly order-bounded processes; the S-integration topology; S-integrable processes; definition of the S-integral; the Riemann-sum integral of u is the S-integral of u-; S-integration is uniformly continuous on uniformly previsibly bounded sets; a dominated convergence theorem; previsible, previsibly order-bounded processes are S-integrable.

646 Basic properties of the S-integral
Splitting a lattice; indefinite S-integrals; martingales and uniformly integrable capped-stake variation sets; change of variable; Itô's formula again.

647 Changing the filtration II
Simultaneously expanding every algebra in a filtration by a single element; controlling [[S-∫u dv≠0]].

649 Pathwise integration
Calculating a Riemann-sum integral one path at a time (i) by Bichteler's construction (ii) with a measure-converging filter; the S-integral for non-decreasing integrators.

648 Changing the algebra II
Subalgebras, induced stochastic integration structures and S-integrals.

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Revised 26.8.2020