Kernels
2024-04-06 09:20:42
Let \(E\) be a set and \(\mathscr{E}\) a \(\sigma\)-algebra of subsets of \(E\). Assume that the
\(\sigma\)-algebra \(\mathscr{E}\) is countably generated, i.e. generated by a countable collection
of subsets of \(E\).
The measurable space \((E, \mathscr{E})\) is called the state space and the
points of \(E\) are called states.
The symbol \(\mathscr{E}\) will also be used to denote the
collection of extended real valued measurable functions on \((E, \mathscr{E})\).
The symbols \(x, y,\ldots\) denote states, \(A, B, \ldots\) denote elements of the \(\sigma\)-algebra \(\mathscr{E}\),
and \(f,g,\ldots\) denote extended real valued measurable functions on \((E, \mathscr{E})\).
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