Comparison Between Several Ways of Defining Lebesgue’s Integral After Lebesgue

Rethinking the Lebesgue Integral

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) ...

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 ...

Fuzzy set-valued stochastic Lebesgue integral

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...