Difference between revisions of "Locally integrable function"
From Encyclopedia of Mathematics
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| − | ''at a point | + | {{TEX|done}} |
| + | ''at a point $M$'' | ||
| − | A function that is integrable in some sense or other in a neighbourhood of | + | 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 [[#References|[2]]]) 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. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.P. Tolstov, "On the curvilinear and iterated integral" ''Trudy Mat. Inst. Steklov.'' , '''35''' (1950) pp. 1–101 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.P. Tolstov, "On the curvilinear and iterated integral" ''Trudy Mat. Inst. Steklov.'' , '''35''' (1950) pp. 1–101 (In Russian)</TD></TR></table> | ||
Latest revision as of 14:23, 17 July 2014
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 [2]) 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.
References
| [1] | S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) |
| [2] | G.P. Tolstov, "On the curvilinear and iterated integral" Trudy Mat. Inst. Steklov. , 35 (1950) pp. 1–101 (In Russian) |
How to Cite This Entry:
Locally integrable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_integrable_function&oldid=16313
Locally integrable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_integrable_function&oldid=16313
This article was adapted from an original article by I.A. Vinogradova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article