2019/06/27 by Talvila, Erik
#26A39 #46B42 #46F10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.11789
An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions F(x,y) defined on the extended real plane [-∞,∞]2 that vanish when x or y is -∞. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative ∂2/(∂ x∂ y) of this space of primitives. If f=∂2/(∂ x∂ y) F then the integral over interval [a,b]× [c,d] ⊆[-∞,∞]2 is ∫ab∫cd f=F(a,c)+F(b,d)-F(a,d)-F(b,c) and ∫-∞^∞ ∫-∞^∞ f=F(∞,∞). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is ‖ f‖=‖ F‖_∞ where F is the unique primitive of f. The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both L1 and the space of Henstock--Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in ‖ ⋅‖_∞ are also inherited by the integrable distributions. It is shown that the dual space are the functions of bounded Hardy--Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder inequality, second mean value theorem, Fubini theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in \mathbb Rn are sketched out.