Measure of unbounded set

In summary: With a different measure this wouldn't necessarily be true (e.g. if your measure assigned infinite measure to every set).In summary, the Lebesgue Outer Measure is a useful tool for analyzing sets, but the possibility of infinite measure for some sets must be taken into account. If A is not a bounded set and m(A∩B)≤(3/4)m(B) for every B, then m(A) can be 0, between 0 and infinity, or infinity, depending on the properties of A and the measure used.
  • #1
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Suppose A is not a bounded set and m(A∩B)≤(3/4)m(B) for every B. what is m(A)??

here, m is Lebesgue Outer Measure

My attemption is :

Let An=A∩[-n,n], then m(A)=lim m(An)= lim m(An∩[-n,n]) ≤ lim (3/4)m([-n,n]) = infinite.

is my solution right? I am confusing m(A) < infinite , it doest make sense for me. Could someone help me?
 
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  • #2
What you wrote is correct as far as it goes, but ##m(A) \leq \infty## doesn't tell you anything new: this is of course true of the outer measure of any set.

Certainly ##m(A) = 0## is possible: consider ##A = \mathbb{Q}##, for example.

Is ##m(A) > 0## possible? Hint: consider ##A = B##.
 
  • #3
There are three possible cases worth thinking about.
- [itex]m(A)=0[/itex], which jbunniii showed is possible.
- [itex]0<m(A)<\infty[/itex], for which jbunniii provided a very useful hint.
- [itex]m(A)=\infty[/itex]... Is this possible? Consider the sets [itex]A_n=A\cap[-n,n][/itex] you defined. If we have to have [itex]0<m(A_n)<\infty[/itex] for some [itex]n\in \mathbb N[/itex] (Is this true?), then maybe the same trick as above can be reused.

It's worth noting that the answer to this question depends on a special property of the Lebesgue measure on [itex]\mathbb R[/itex], which fails for some other infinite measures. Namely, we're using the property that the whole space is a countable union of finite-measure sets.
 

What is a measure of unbounded set?

A measure of an unbounded set is a numerical value that represents the size or extent of the set. It is used to quantify the amount of elements or space within the set.

What is the difference between bounded and unbounded sets?

Bounded sets have a finite size, meaning they have a defined limit or boundary. Unbounded sets, on the other hand, have an infinite size and do not have a defined limit or boundary.

How is a measure of unbounded set calculated?

The measure of an unbounded set can be calculated in several ways, depending on the specific set. For example, the measure of a geometric unbounded set (such as a line or plane) can be calculated using geometric formulas, while the measure of an unbounded data set can be calculated using statistical methods.

Why is it important to measure unbounded sets?

Measuring unbounded sets allows us to understand and analyze the size and distribution of elements within the set. This can provide valuable insights and help us make informed decisions in various fields, such as mathematics, physics, and data analysis.

What are some examples of unbounded sets?

Examples of unbounded sets include the set of all real numbers, the set of all positive integers, and the set of all possible outcomes in a probability experiment. In geometry, a line, plane, or three-dimensional space can also be considered unbounded sets.

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