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 find a condition for a Borel mapping $f:{ℝ}^m\to {ℝ}^n$ which implies that the Hausdorff dimension of $f^{-1}\left(y\right)$ is less than or equal to m-n for almost all $y\in {ℝ}^n$.

A weak form of the Henstock Lemma for the $\mathrm{P}oU$-integrable functions is given. This allows to prove the existence of a scalar Volterra derivative for the $\mathrm{P}oU$-integral. Also the $\mathrm{P}oU$-integrable functions are characterized by means of Pettis integrability and a condition involving finite pseudopartitions.

Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_{\varphi }$ of Wiener measure and for a partition $0=t_0<t_1<\cdots <t_n<t_{n+1}=T$ of $[0,T]$, let $X_{n}C[0,T]\to {ℝ}^{n+1}$ and $X_{n+1}C[0,T]\to {ℝ}^{n+2}$ be given by $X_n\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_n\right))$ and $X_{n+1}\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_{n+1}\right))$, respectively. In this paper, using a simple formula for the conditional $w_{\varphi }$-integral of functions on $C[0,T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_{\varphi }$-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_{\varphi }$-integral...

Let $T$ be a locally compact Hausdorff space and let $C_0\left(T\right)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma$-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u{C}_0\left(T\right)\to X$ when...

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.