We give an extension of de Finetti’s concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to unbounded random variables that we believe has advantages over Lebesgue-style integrals in the finitely additive setting. We also give a general version of th...

The Fourier transform is considered as a Henstock–Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann–Lebesgue lemma fails: Henstock– Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appen...

Abstract. We show that an anticipating stochastic forward integral introduced in [8] by means of fractional calculus is an extension of other forward integrals known from the literature. The latter provide important classes of integrable processes. In particular, we investigate the deterministic case for integrands and integrators from optimal Besov spaces. Here the forward integral agrees with...

For an arbitrary innnite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly diierentiable; thus, for these functions the Lebesgue Diierentiation Theorem fails rather spectacularly. We also relate the degree of nondiierentiability of the indeenite Pettis integral to the cotype of X, fr...

In 1986 Bruckner, Fleissner and Foran [2] obtained a descriptive definition of a minimal extension of the Lebesgue integral which integrates the derivative of any differentiable function. Recently, Bongiorno, Di Piazza and Preiss [1] showed that this minimal integral can be obtained from McShane’s definition of the Lebesgue integral [4] by imposing a mild regularity condition on McShane’s parti...

The link between fractional and stochastic calculus established in part I of this paper is investigated in more detail. We study a fractional integral operator extending the Lebesgue–Stieltjes integral and introduce a related concept of stochastic integral which is similar to the so–called forward integral in stochastic integration theory. The results are applied to ODE driven by fractal functi...

A basic theme in the wonderful books and surveys of Stein, Weiss, and Zygmund is that Hilbert transforms, Poisson kernels, heat kernels, and related objects are quite interesting and fundamental. I certainly like this point of view. There is a variety of ways in which things can be interesting or fundamental, of course. In the last several years there have been striking developments connected t...

in which the functions r, p and q are continuous on (0, °o) while r is positive there. The point x = 0 is singular for the functional in the sense that the conditions on r, p, and q may not hold for an interval of the form [O, b]. Finally, all integrals which appear are Lebesgue integrals. We denote by: F[0, b], the class of all functions y such that y is absolutely continuous and y'ÇzL2 on eve...

For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, fr...

We prove L p $L^{p}$ -boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate convergence partial Fourier integrals vector-valued functions. Our proof relies bounds wave packet embeddings into outer Lebesgue spaces time-frequency-scale space R + 3 $\mathbb {R}^{3}_{+}$ , which are focus this paper.