Laplace Functional

The Laplace functional of a random measure is the functional

where denotes the expected value; the functional is defined for suitable nonnegative measurable functions . It is called a functional because its argument is the entire function , rather than a number. The Laplace functional generalizes the Laplace transform of a random variable and, under standard conditions, determines the probability law of the random measure. Given a sigma-algebra representing available information, the conditional Laplace functional is the conditional expectation

It is therefore a random quantity determined by that information; conditioning on an initial state or on an event gives common special cases.