The minimal integral which includes Lebesgue integrable functions and derivatives

Rethinking the Lebesgue Integral

Which Powers of Holomorphic Functions Are Integrable?

Question 1. Let f(z1, . . . , zn) be a holomorphic function on an open set U ⊂ C. For which t ∈ R is |f |t locally integrable? The positive values of t pose no problems, for these |f |t is even continuous. If f is nowhere zero on U then again |f |t is continuous for any t ∈ R. Thus the question is only interesting near the zeros of f and for negative values of t. More generally, if h is an inve...

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...

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 ...

On linearity of pan-integral and pan-integrable functions space

Article history: Received 14 June 2017 Received in revised form 8 August 2017 Accepted 8 August 2017 Available online xxxx