### A Fubini-type theorem for finitely additive measure spaces

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y → [0,∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel’s perturbation technique.

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y → [0,∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic...

We prove that for λ ∈ [0,1] and A, B two Borel sets in ${\mathbb{R}}^{n}$ with A convex, ${\Phi}^{-1}\left({\gamma}_{n}(\lambda A+(1-\lambda )B)\right)\ge \lambda {\Phi}^{-1}\left({\gamma}_{n}\left(A\right)\right)+(1-\lambda ){\Phi}^{-1}\left({\gamma}_{n}\left(B\right)\right)$, where ${\gamma}_{n}$ is the canonical gaussian measure in ${\mathbb{R}}^{n}$ and ${\Phi}^{-1}$ is the inverse of the gaussian distribution function.

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.

Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series ${\sum}_{n=1}^{\infty}{c}_{n}f\left({T}^{n}x\right)$ converges almost everywhere with respect to Lebesgue measure provided that ${\sum}_{n=1}^{\infty}{\left|{c}_{n}\right|}^{2}lo{g}^{2}n<\infty $.

Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.