In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure.
Definition Let ζ {\displaystyle \zeta } be a random measure on the measurable space ( S , A ) {\displaystyle (S,{\mathcal {A}})} and denote the expected value of a random element Y {\displaystyle Y} with E [ Y ] {\displaystyle \operatorname {E} [Y]} . The intensity measure
E ζ : A → [ 0 , ∞ ] {\displaystyle \operatorname {E} \zeta \colon {\mathcal {A}}\to [0,\infty ]}
of ζ {\displaystyle \zeta } is defined as
E ζ ( A ) = E [ ζ ( A ) ] {\displaystyle \operatorname {E} \zeta (A)=\operatorname {E} [\zeta (A)]}
for all A ∈ A {\displaystyle A\in {\mathcal {A}}} . Note the difference in notation between the expectation value of a random element Y {\displaystyle Y} , denoted by E [ Y ] {\displaystyle \operatorname {E} [Y]} and the intensity measure of the random measure ζ {\displaystyle \zeta } , denoted by E ζ {\displaystyle \operatorname {E} \zeta } .
Properties The intensity measure E ζ {\displaystyle \operatorname {E} \zeta } is always s-finite and satisfies
E [ ∫ f ( x ) ζ ( d x ) ] = ∫ f ( x ) E ζ ( d x ) {\displaystyle \operatorname {E} \left[\int f(x)\;\zeta (\mathrm {d} x)\right]=\int f(x)\operatorname {E} \zeta (dx)}
for every positive measurable function f {\displaystyle f} on ( S , A ) {\displaystyle (S,{\mathcal {A}})} .
See also Intensity (measure theory)
References
