Vol. 34, No. 3, 1970

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A generalized Radon-Nikodym derivative

Hugh D. Brunk and Søren Glud Johansen

Vol. 34 (1970), No. 3, 585–617

Let {νa,a R} be a family of signed measures on a σ-field 𝒜 of subsets of an abstract space Ω. Let be a sub σlattice of 𝒜. Under certain conditions we associate with the family of measures and a function f, which we call the Lebesgue-Radon-Nikodym (LRN) function. The function f is measurable 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.

Mathematical Subject Classification
Primary: 28.16
Secondary: 60.00
Received: 8 May 1969
Published: 1 September 1970
Hugh D. Brunk
Søren Glud Johansen