The Lebesgue integral as a Riemann integral

The Choquet integral as Lebesgue integral and related inequalities

The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee ...

Rethinking the Lebesgue Integral

Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral; application to the calculus of ∆-antiderivatives

The theory of dynamic equations appears in the literature in 1988 in the Ph. D. of S. Hilger [18]. The aim of this theory consists in to study differential and difference equations under the same formulation. Thus, the concepts of ∆– derivative and ∆−integral have been defined. These two concepts cover both the classical derivative and the Riemann integral together with the forward and sum oper...

Notes on the Lebesgue Integral

In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined in terms of limits of the Riemann sums ∑n−1 j=0 f(x ∗ j)∆j, where ∆j = xj+1− xj. The basic idea for the Lebesgue integral is to partition the y-axis, which contains the range of f , rather than the x-axis. This seems like a “dumb” idea at first. Shouldn’t the two ways end up givin...

Lebesgue Integral of Simple Valued Function

The following propositions are true: (1) Let n, m be natural numbers, a be a function from [: Segn, Segm :] into R, and p, q be finite sequences of elements of R. Suppose that (i) dom p = Seg n, (ii) for every natural number i such that i ∈ dom p there exists a finite sequence r of elements of R such that dom r = Segm and p(i) = ∑ r and for every natural number j such that j ∈ dom r holds r(j) ...