Locally integrable function
at a point $M$
A function that is integrable in some sense or other in a neighbourhood of $M$. If a real-valued function $f$, defined on the interval $[a,b]$, is the pointwise finite derivative of a function $F$, real-valued and defined on this interval, then $f$ is locally Lebesgue integrable at the points of an open everywhere-dense set on $[a,b]$. In the two-dimensional case (see ) there is a real-valued function $f$, defined on the square $[0,1]\times[0,1]$, that is the pointwise finite mixed derivative in either order $\partial^2F/\partial x\partial y=\partial^2F/\partial y\partial x=f(x,y)$ and that is not locally Lebesgue integrable at any point of the square.
|||S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)|
|||G.P. Tolstov, "On the curvilinear and iterated integral" Trudy Mat. Inst. Steklov. , 35 (1950) pp. 1–101 (In Russian)|
Locally integrable function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Locally_integrable_function&oldid=32462