### A convolution inequality concerning Cantor-Lebesgue measures.

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A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from . The theory is recovered together with a few new results.

In this paper, we define a $$-integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $\mu $ is compatible with its topology in the sense that every open set is $\mu $-measurable. We prove that the $$-integral is equivalent to $\mu $-integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.

We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space ${L}^{p}(\mu ,X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of ${L}^{p}(\mu ,X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological space and...