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Let {νa,a ∈ R} be a family
of signed measures on a σ-field 𝒜 of subsets of an abstract space Ω. Let ℳ be a sub
σ− lattice of 𝒜. Under certain conditions we associate with the family of measures
and ℳ a function f, which we call the Lebesgue-Radon-Nikodym (LRN) function.
The function f is measurable ℳ and satisfies the relations
| νa(B ∩ [f < a]) ≦ 0, | a ∈ R, B ∈ℳ, | |
| | νb(C ∩ [f > b]) ≧ 0, | b ∈ R, C ∈ℳc. | | |
This paper contains a construction of f by means of a JordanHahn decomposition
for σ-lattices, and gives various characterizations and representations of
f.
Special cases are: the derivative of a signed measure with respect to a nonnegative
measure, conditional expectation given a σ-field, and conditional expectation given a
σ-lattice. The LRN function also provides a conditional generalized mean whose
relationship to the generalized mean parallels the relationship of the conditional
expectation to the expectation.
The paper also contains a convergence theorem for LRN functions with respect to
an increasing sequence of σ-lattices, thus generalizing the martingale convergence
theorem.
Finally it is proved that f is the solution to a minimization problem,
generalizing known minimizing properties of conditional expectation and
of conditional expectation given a σ-lattice. These properties exhibit the
latter as solution of various problems of restricted maximum likelihood
estimation.
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