On the Fundamental Existence Theorem of Kishi

Source abstract

Let Ω be a locally compact Hausdorff space and G(x, y) be a strictly positive lower semicontinuous function on the product space Ω×Ω of Ω . Such a function G(x, y) is called a kernel on Ω . The adjoint kernel Ğ(x, y) of G(x, y) is defined by Ğ(x, y) =G(y, x) . Whenever we say a measure on Ω , we mean a positive regular Borel measure on Ω . The potential G μ (x) and the adjoint potential Ğ μ (x) of a measure μ relative to the kernel G(x, y) is defined by respectively. These are also strictly positive lower semicontinuous functions on Ω provided μ≠0.