The ideal probabilistic forecast for a random variable Y based on an information set F is the conditional distribution of given F. In context point forecasts aiming to specify functional T such as mean, quantile or risk measure, respective applied distribution. This paper provides theoretical justification why this actually forecast, that is, F-measurable variable. To end, appropriate notion me...

For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two...

The lower dimension $\dim_L$ is the dual concept of Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced modified $\dim_{ML}$ by making monotonic with simple formula $\dim_{ML} X=\sup\{\dim_L E: E\subset X\}$. our first result we prove that finitely stable in any metric space, answering a question Yu. We new, characterization for For space $X$ let $\mathcal{K}(X)$ denote non...

In this paper we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems.