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A polar representation of a complex measure mu is analogous to the polar representation of a complex number as z=re^(itheta), where r=|z|, dmu=e^(itheta)d|mu|. (1) The analog ...

Two complex measures mu and nu on a measure space X, are mutually singular if they are supported on different subsets. More precisely, X=A union B where A and B are two ...

Let f be a real-valued function defined on an interval [a,b] and let x_0 in (a,b). The four one-sided limits D^+f(x_0)=lim sup_(x->x_0+)(f(x)-f(x_0))/(x-x_0), (1) ...

Measure theory is the study of measures. It generalizes the intuitive notions of length, area, and volume. The earliest and most important examples are Jordan measure and ...

A generalization of the Lebesgue integral. A measurable function f(x) is called A-integrable over the closed interval [a,b] if m{x:|f(x)|>n}=O(n^(-1)), (1) where m is the ...

Let f(theta) be Lebesgue integrable and let f(r,theta)=1/(2pi)int_(-pi)^pif(t)(1-r^2)/(1-2rcos(t-theta)+r^2)dt (1) be the corresponding Poisson integral. Then almost ...

A positive measure is a measure which is a function from the measurable sets of a measure space to the nonnegative real numbers. Sometimes, this is what is meant by measure, ...

The class of continuous functions is called the Baire class 0. For each n, the functions that can be considered as pointwise limits of sequences of functions of Baire class ...

Any locally compact Hausdorff topological group has a unique (up to scalars) nonzero left invariant measure which is finite on compact sets. If the group is Abelian or ...

Let T be a linear operator on a separable Hilbert space. The spectrum sigma(T) of T is the set of lambda such that (T-lambdaI) is not invertible on all of the Hilbert space, ...