On the Fundamental Existence Theorem of Kishi

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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 Gμ(x) and the adjoint potential Ğμ(x) of a measure μ relative to the kernel G(x, y) is defined byrespectively. These are also strictly positive lower semicontinuous functions on Ω provided μ≠0.

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