Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Topology and Analysis
N-dimensional Lebesgue measure: def. with Borel sets
Reply to thread
Message
[QUOTE="DavideGenoa, post: 5502020, member: 484525"] Let us define, as Kolmogorov-Fomin's [I]Элементы теории функций и функционального анализа[/I] does, the definition of [I]outer measure[/I] of a bounded set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of ##A## by finite or countable families of ##n##-parallelipeds ##P_k=\prod_{i=1}^n I_i##, where ##I_i\subset\mathbb{R}## are finite intervals (where one point may be considered an interval), whose measure ##m(P_k)## is the product of the lengths of the intervals #I_i#. A set is said to be [I]elementary[/I] when it is the union of a finite number of such ##n##-parallelipeds and a set ##A## is said to be *measurable* if, for any ##\varepsilon>0##, there is an elementary set ##B## such that $$\mu^{\ast}(A\triangle B)<\varepsilon.$$The function ##\mu^{\ast}## defined on measurable sets only is called [I]Lebesgue measure[/I] and the notation ##\mu## is used for it. I have been told that, in the definitions explained above, we can equivalently use Borel sets where ##n##-paralallelepipeds appear. How can we prove it? Let us use the index ##B## for such alternative definitions. It is clear to me that, for any set ##A\subset \mathbb{R}^n##, $$\mu_B^{\ast}(A)\le \mu^{\ast}(A)$$ because ##n##-parallelepipeds are Borel sets, and if ##A## is measurable according to the ##n##-parallelepipeds-based definition, it also is according to the Borel-sets-based definition, but I cannot prove the converse... [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Topology and Analysis
N-dimensional Lebesgue measure: def. with Borel sets
Back
Top