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To0ta
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let [tex]\mu^{}*[/tex] , v[tex]^{}*[/tex] outer measura on X . Show that max{[tex]\mu^{}*[/tex] , v[tex]^{}*[/tex]} is an outer measure on X ?
Tinyboss said:The first (possibly only) thing to try would be to look at the properties that define an outer measure, and check whether max(u,v) satisfies them. Did you try that yet?
And what was the result?To0ta said:Yes I have tried a lot
jgm340 said:FAIL.
From wikipedia: http://en.wikipedia.org/wiki/Outer_measure#Formal_definitions
Defining properties of an outer measure:
* The empty set has measure 0.
* Monotonicity: If A is a subset of B, then the measure of A is at most the measure of B.
* Countable Subadditivity: The measure of a countable union of sets is at most the sum of the measures of each of the sets in the union.
If u* and v* are outer measures, then max{u*,v*} is outer measure if and only if it satisfies the above three properties.
In other words:
* max{u*(empty set), v*(empty set)} = 0.
* If A is a subset of B, then max{u*A,v*A} is less than or equal to max{u*B,v*B}.
*If A1, A2, A3, ... are sets, and A is their union, then max{u*A,v*A} is less than or equal to the sum over i = 1,2,3,... of max{u*Ai,v*Ai}.
The first two conditions are really, really straightforward. The third follows from the fact that the maximum of two sums (say, for example, of max{sum of x_i, sum of y_i}) is at most the sum of the maximums (i.e. the sum of max{x_i,y_i}).
Lebesgue measurable sets are a type of set used in measure theory, a branch of mathematics that deals with the concept of size or measure of sets. They were introduced by French mathematician Henri Lebesgue in the early 20th century as an extension of the more basic concept of "measurable sets" defined by his mentor, mathematician Émile Borel. Lebesgue measurable sets have important applications in areas such as probability theory, functional analysis, and mathematical physics.
Unlike other types of measurable sets, Lebesgue measurable sets have the property that their measure or size can be defined in a consistent way for any set, regardless of its shape or structure. This is known as the "Lebesgue measure property" and is a key feature that distinguishes Lebesgue measurable sets from other types of measurable sets, such as Jordan measurable sets or Borel measurable sets.
Lebesgue measurable sets are a fundamental concept in measure theory, providing a framework for defining and calculating the size or measure of sets in a consistent and rigorous manner. They also allow for the development of more advanced concepts and techniques in measure theory, such as Lebesgue integration and the Lebesgue differentiation theorem.
Lebesgue measurable sets are defined as sets that satisfy the Lebesgue measure property, which states that for any set, the measure or size is equal to the infimum (greatest lower bound) of the measures of all "outer approximations" of the set. This means that the measure of a set is equal to the smallest possible measure that can be obtained by enclosing the set with a sequence of simpler sets (such as intervals or rectangles). Lebesgue measurable sets can also be characterized as sets that can be approximated from the outside by open sets and from the inside by closed sets.
Some examples of Lebesgue measurable sets include intervals and rectangles in one, two, or three dimensions, as well as more complex sets such as the Cantor set and the Sierpinski triangle. In general, any set that can be approximated from the outside by open sets and from the inside by closed sets can be considered Lebesgue measurable. However, there are also sets that are not Lebesgue measurable, such as the Vitali set, which cannot be approximated in a consistent manner by simpler sets.