Functions in Lploc[0,∞) where 1 ≤ p ≤ ∞ can be considered as inputs to linear systems. However, hysteresis operators of Preisach type have only been defined on much smaller space of regulated (or Baire) functions. In this paper, we re-define Play operators so that they are well defined for real valued measurable functions. We show that this definition coincides with the older definition for con...

For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, fr...

It is well known that the disconnectedness of a non-Archimedean totally ordered field in the order topology makes integration more difficult than in the real case. In this paper, we present a remedy to that difficulty and study measure theory and integration on the Levi-Civita field. After reviewing basic elements of calculus on the field, we introduce a measure that proves to be a natural gene...

All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all rotation spherical also

We prove that two basic questions on outer measure are undecid-able. First we show that consistently • every sup-measurable function f : R 2 −→ R is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functions f : R 2 −→ R the Cauchy problem y ′ = f (x, y), y(x 0) = y 0 has a unique almost-everywhere solution in the class AC l (R) o...