Suppose we consider Banach sp valued mappings and integration in several variables.
1) in general one can also define refinement integral . i m also aware that it does not make difference to hk integral even for banach space valued maps.
2) but Riemann integral of two variables is defined on,ly using NEt partttions. for real functions abolute integrability will coincide with lebesgue and we have only one unique daniell extenmsion.
but Suppose i start with normed space of riemann integrable mappings ( not functions) then riemann integral is not absolute integral for mappings and we can extend by a method which is variant of daniell process described by professor M.Leinert.
we can also extend integral on step maps .
we can also extend hk integral for maps
1) is the daniell extension of riemann integral different from hk integral in the sense there are are mappings which are daniellriemann integrable but not hk?
2) it is well known that if we restrict our attention to only net parttions then given agauge delta ( a rectangle containing a point) one need not have a delta fine net partition.
3) however if we consider gauges as product of gauges on x and y axis then it will exist . why not use such gauges
4) the only obstacle seems to be that if a set is of measure zero( def : integral of characteristic function is zero is a Null set ( def : integral of any positive function is zero on that set.) The result may not hold good.
5) but then we use only null sets. and we have each negilgible set ( def one which can be covered by countable union of rectangles so that sum total volume is arbitrarily small ) is a null set..
i shall ve very happy if discussion and guidance on these issues is available. i may post on research gate as well.
6) so can we use such gauges ?
yes very intricate questions .
all knowledgable persons are invited to join share and consider these open problems in very basic analysis
Many questions in one problem. I suggest first to study real-valued case. Answers are not so simple. If you are interested in Daniell's approach: Bianconi, Ricardo, João C. Prandini, and Cláudio Possani. "A Daniell integral approach to nonstandard Kurzweil-Henstock integral." Czechoslovak Mathematical Journal 49.4 (1999): 817-823.
For a general definition of HK integrals defined on sets in R^n: please check some papers by the authors of https://www.researchgate.net/publication/242012453_An_integral_defined_by_approximating_BV_partitions_of_unity
Yours,
M.C.
Article An integral defined by approximating BV partitions of unity
i know answers for real valued cases. i know full theory of hk integral. i fact i have set up a complete axiomatic theory .
since it is single question let me make things clear.
any daniell extension is unique for functions on real line . so question is resolved ffor functions. i know the answer.
so many subquestions help to clarify the question
1) Henstock-kurzweil integral is charcterized by using delta fine tagged parttions for a gauage . a gauge delta on real line is a positive real number ( or nondegerate iclosed interval) a gauge in two variables is a closed nondegenrate rctangle containg the poin
2) Riemann integral uses ordinary partitions of bounded rectangles. and is defined using refinement. it has been answered positively that Henstock-kurzweil integral includes riemann integral for Banach space valued maps defined using refinement.
3) When we go to two variables the theorem will work provided riemann integral is defined using all parttions
4) to define Henstock-kurzweil integral one needs to assert existence of delta fine tagged partttioions of a gauge.. as shown by a counter example if one uses only net partitions then delta fine tagged partition may not exist. so essentially one has to use bipartitions.
5) herein lies the Hitch .Riemann integral is defined using only net parttions. so set of admissible partitions is smaller but one can not define Henstock-Kurzweil integral using these partitions. as explained above.
6) of course for functions if one carries an extension of rieamnn integralcarry out daniell extension since it is unique we have lebesgue integral and henstock-kurzweil integral includes lebesgue integral . so we are clear. the answer is known to us
6) .but for banach space valued mappings if one carries an extension of Riemann intgeral ( it is non absolute) by daniell-Leinert method and carry an extension of henstock-kurzweil integral we do not know the answer.
Lebesgue integral is included in hk case.
7) now stems another branch if we consider the collection of gauges which is obtained as product of two gauges on two axes then one can define gauge integral only using net partitions as delta fine tagged partitions exist .
all thorems can be proved nincluding fubini. stokes also.
8)but there is a hitch. the theorem each set of measure zero is a null set can not be proved.
a set is of measure zero if integral of indicator functionn is zero.
a set is null set if integral of any positive function is zero.
9)the standard proof fails as we shrink gauge a countable number of times. may be the same gauge on x axis will occur for shrinking if we use product gauges and proof fails.
10) we have a way out a set is aclled negligible if it can be covered by countable union of recxatnagles with total rea aribtary small. a negligible set is a null set.
11) so we stick to null sets and develop theory this integral i think will have all theorems ?
i hope i have tried to raise basic questions in a fundamental branch of maths. we do not want any unnatural concepts like bv integral and nonstandard analysis. basic bbut deep question
it seems that elegant yet simple revolutionary paper by M.A Robdera "unified approach to vector integration " internet journal of operator thgeory functional analysis and applications vol 5, number 2, 2013 pp119-139 pushpa publications , allahabad india contains key to the answers to my questionset
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