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| Preface | |
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| Vector Integration | |
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| Preliminaries | |
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| Banach spaces | |
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| Classes of sets | |
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| Measurable functions | |
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| Simple measurability of operator-valued functions | |
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| Weak measurability | |
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| Integral of step functions | |
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| Totally measurable functions and the immediate integral | |
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| The Riesz representation theorem | |
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| The classical integral | |
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| The Bochner integral | |
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| Convergence theorems | |
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| Measures with finite variation | |
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| The variation of vector measures | |
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| Boundedness of [sigma]-additive measures | |
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| Variation of real-valued measures | |
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| Integration with respect to vector measures with finite variation | |
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| The indefinite integral | |
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| Integration with respect to gm | |
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| The Radon-Nikodym theorem | |
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| Conditional expectations | |
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| [sigma]-additive measures | |
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| [sigma]-additive measures on [sigma]-rings | |
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| Uniform [sigma]-additivity | |
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| Uniform absolute continuity and uniform [sigma]-additivity | |
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| Weak [sigma]-additivity | |
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| Uniform [sigma]-additivity of indefinite integrals | |
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| Weakly compact sets in L[superscript 1 subscript F] ([mu]) | |
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| Measures with finite semivariation | |
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| The semivariation | |
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| Semivariation and norming spaces | |
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| The semivariation of [sigma]-additive measures | |
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| The family m[subscript F,Z] of measures | |
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| Integration with respect to a measure with finite semivariation | |
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| Measurability with respect to a vector measure | |
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| The seminorm m[subscript F,G](f) | |
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| The space of integrable functions | |
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| The integral | |
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| Convergence theorems | |
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| Properties of the space F[subscript D] (B, m[subscript F,G]) | |
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| Relationship between the spaces F[subscript D](m) | |
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| The indefinite integral of measures with finite semivariation | |
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| Strong additivity | |
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| Extension of measures | |
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| Applications | |
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| The Riesz representation theorem | |
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| Integral representation of continuous linear operations on L[superscript p]-spaces | |
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| Random Gaussian measures | |
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| The Stochastic Integral | |
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| Summable processes | |
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| Notations | |
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| The measure I[subscript X] | |
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| Summable processes | |
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| Computation of I[subscript X] for predictable rectangles | |
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| Computation of I[subscript X] for stochastic intervals | |
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| The stochastic integral | |
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| The space F[subscript D] [characters not reproducible] | |
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| The integral [function of] HdI[subscript X] | |
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| A convergence theorem | |
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| The stochastic integral H - X | |
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| The stochastic integral and stopping times | |
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| Stochastic integral of elementary processes | |
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| Stopping the stochastic integral | |
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| Summability of stopped processes | |
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| The jumps of the stochastic integral | |
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| Convergence theorems | |
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| The completeness of the space L[superscript 1 subscript F,G](X) | |
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| The Uniform Convergence Theorem | |
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| The Vitali and the Lebesgue Convergence Theorems | |
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| The stochastic integral of [sigma]-elementary and of caglad processes as a pathwise Stieltjes integral | |
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| Summability of the stochastic integral | |
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| Summability criterion | |
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| Quasimartingales and the Doleans measure | |
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| The summability criterion | |
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| Local summability and local integrability | |
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| Definitions | |
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| Basic properties | |
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| Convergence theorems | |
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| Additional properties | |
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| Martingales | |
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| Stochastic integral of martingales | |
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| Square integrable martingales | |
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| Extension of the measure I[subscript M] | |
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| Summability of square integrable martingales | |
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| Properties of the space F[subscript F,G](M) | |
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| Isometrical isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]]) | |
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| Processes with Finite Variation | |
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| Functions with finite variation and their Stieltjes integral | |
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| Functions with finite variation | |
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| The variation function g | |
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| The measure associated to a function | |
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| The Stieltjes integral | |
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| Processes with finite variation | |
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| Definition and properties | |
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| Optional and predictable measures | |
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| The measure [mu subscript X] | |
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| Summability of processes with integrable variation | |
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| The stochastic integral as a Stieltjes integral | |
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| The pathwise stochastic integral | |
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| Semilocally summable processes | |
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| Processes with Finite Semivariation | |
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| Functions with finite semivariation and their Stieltjes integral | |
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| Functions with finite semivariation | |
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| Semivariation and norming spaces | |
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| The measure associated to a function | |
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| The Stieltjes integral with respect to a function with finite semivariation | |
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| Processes with finite semivariation | |
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| The semivariation process | |
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| The measure [mu]x | |
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| Summability of processes with integrable semivariation | |
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| The pathwise stochastic integral | |
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| Dual projections | |
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| Dual projection of measures | |
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| Dual projections of processes | |
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| Existence of dual projections | |
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| Processes with locally integrable variation or semivariation | |
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| Examples of processes with locally integrable variation or semivariation | |
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| Decomposition of local martingales | |
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| The Ito Formula | |
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| The Ito formula | |
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| Preliminary results | |
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| The vector quadratic variation | |
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| The quadratic variation | |
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| The process of jumps | |
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| Ito's formula | |
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| Stochastic Integration in the Plane | |
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| Preliminaries | |
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| Order relation in R[superscript 2] | |
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| The increment [Delta subscript zz], g | |
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| Right continuity | |
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| The filtration | |
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| The predictable [Sigma]-algebra | |
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| Stopping times | |
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| Stochastic processes | |
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| Extension of processes from R[superscript 2 subscript +] [times] [Omega] to R[superscript 2] [times] [Omega] | |
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| Summable processes | |
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| The measure I[subscript X] | |
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| Summable processes | |
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| The seminorm I[subscript X] and the space F[subscript F,G](X) | |
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| The integral [function of] HdI[subscript X] | |
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| The stochastic integral H - X | |
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| Properties of the stochastic integral | |
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| Convergence theorems | |
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| Extension of I[subscript X] to P([infinity]) | |
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| Existence of left limits of X in L[superscript p subscript E] | |
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| Some properties of the integral [function of] HdI[subscript X] | |
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| Summability of stopped processes | |
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| Summability of the stochastic integral | |
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| Two-Parameter Martingales | |
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| Martingales | |
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| Square integrable martingales | |
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| A decomposition theorem | |
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| The measures [characters not reproducible] and [mu subscript [M]] | |
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| Summability of the square integrable martingales in Hilbert spaces | |
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| The space F[subscript F,G](I[subscript M]) | |
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| Isometric isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]]) | |
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| Two-Parameter Processes with Finite Variation | |
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| Functions with finite variation in the plane | |
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| Monotone functions | |
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| Partitions | |
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| Variation corresponding to a partition | |
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| Variation of a function on a rectangle | |
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| Limits of the variation | |
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| The variation function g | |
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| Functions with finite variation | |
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| Functions vanishing outside a quadrant | |
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| Variation of real-valued functions | |
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| Lateral limits | |
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| Measures associated to functions | |
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| [sigma]-additivity of the measure m[subscript g] | |
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| The Stieltjes integral | |
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| Processes with finite variation | |
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| Processes with integrable variation | |
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| The measure [mu subscript X] | |
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| Summability of processes with integrable variation | |
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| The stochastic integral as a Stieltjes integral | |
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| Two-Parameter Processes with Finite Semivariation | |
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| Functions with finite semivariation in the plane | |
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| Functions with finite semivariation | |
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| Semivariation and norming spaces | |
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| The measure associated to a function | |
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| The Stieltjes integral for functions with finite semivariation in R[superscript 2] | |
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| Processes with finite semivariation in the plane | |
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| Processes with finite semivariation | |
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| The measure [mu subscript X] | |
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| Summability of processes with integrable semivariation | |
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| References | |