Monotone nonincreasing random fields on posets. I

For an arbitrary poset $H$ and measure $\rho$ on $H\times{\mathbf R}$ (where $\mathbf R$ is the real axis), we construct a monotone decreasing stochastic field $\eta_\rho$ and calculate finite-dimensional distributions of the field. In the case where $H$ is a $\wedge$-semilattice and the measure $\rho$ satisfies additional conditions, we calculate characteristics of the field $\eta_\rho$ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field.
The described construction for random fields gives a new method for constructing positively defined functions on posets.