measure zero in Rn∗
asteroid†
2013-03-22 0:47:52
0.1
Measure Zero in Rn
In Rn with the Lebesgue measure m there is a simple characterization of the
sets that have measure zero.
Theorem - A subset X ⊆ Rn has zero Lebesgue measure if and only if for
every > 0Pthere is a sequence of compact rectangles {Ri }i∈N that cover X and
such that i m(Ri ) < .
0.2
Measure Zero Avoiding Measure Theory
In some circumstances one may want to avoid the whole theory of Lebesgue
measure and integration and still be interested in having a notion of measure
zero, like for example when studying Riemann integrals or constructing the
Lebesgue measure from an historical point of view. Another interesting example
where sets of measure zero arise and there is no reason to introduce measures or
integrals is when studying the type of sets where a function of bounded variation
is not differentiable (this sets have always measure zero).
Nevertheless, the notion of measure zero is not lost in this situation. Since
the Lebesgue measure of compact rectangles can be easily calculated and defined
from the start (see Jordan content of an n-cell for example), the condition stated
in the previous theorem can be taken as the definition of measure zero.
Definition - A set X in Rn is said to have measure zero if for every > 0
there
is a sequence of compact rectangles {Ri }i∈N that cover X and such that
P
m(R
i ) < , where m is the Jordan content.
i
Similarly, the notion of almost everywhere remains essentially the same. One
just has to work with the previous definition.
∗ hMeasureZeroInmathbbRni
created: h2013-03-2i by: hasteroidi version: h40452i Privacy
setting: h1i hTheoremi h28A05i
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