Contents of Measure Theory, by D.H.Fremlin

Chapter 61: The Riemann-sum integral

611 Stopping times
Filtrations; the lattice T of stopping times; the regions [[σ<τ]]; suprema and infima in T; the algebra defined by a stopping time; stopping time intervals; enumerating the cells of a finite sublattice of T; covered envelopes and covering ideals.

612 Fully adapted processes
L0 spaces; f-algebras of fully adapted processes; the identity process; progressively measurable classical processes; actions of Borel measurable real functions; simple processes; order-bounded processes; extension of processes to covered envelopes; Brownian motion; the Poisson process.

613 Definition of the integral
Convergence in measure; interval functions; Δe(u,dψ), Riemann sums SI(u,dψ), integrals ∫Su dψ and ∫Su dv; invariance under change of law; integration and covered envelopes; integrating a simple process; indefinite integrals.

614 Moderately oscillatory processes
The ucp topology on Mob; moderately oscillatory processes; approximating a moderately oscillatory process; near-simple processes; classical processes with càdlàg sample paths.

615 Integrators
Capped-stake variation sets; integrators; integration and the ucp topology; if u is moderately oscillatory and v is an integrator, ∫u dv is defined; processes of bounded variation are integrators.

616 Integral identities and quadratic variations
Indefinite integrals are integrators; integrating with respect to an indefinite integral; covariation and quadratic variation; integrating with respect to a covariation; calculating the quadratic variation of the identity process and the Poisson process.

617 Jump-free processes
Oscillations; jump-free processes; classical processes with continous sample paths.

618 Itô's formula
Itô's formula in one dimension; k-tuples of processes; Itô's formula in k dimensions.

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Revised 4.3.2018