Trying to find the measure of a set.

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Discussion Overview

The discussion revolves around the Lebesgue measure, specifically focusing on the measurability and measure of two sets defined in the context of decimal expansions within the interval [0,1). Participants explore the definitions and properties of these sets, seeking to understand how they relate to Lebesgue measurability and null sets.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks to determine the Lebesgue measurability and measure of sets A and B, defined by specific conditions on decimal expansions.
  • Another participant claims to have figured out the measure of set A but expresses uncertainty regarding set B.
  • A different participant asserts that the measure of set B is 0, suggesting that for any finite number of digits being 7, the measure can be expressed as (1/10)^n as n approaches infinity.
  • One participant emphasizes that proving set B is measurable requires expressing it as a union of a Borel set and a null set.
  • Another participant argues that if a set is null, it is measurable, and one can union it with the empty set to satisfy the requirement of having a Borel set.
  • A later reply discusses the countability of finite subsets of a countable set, suggesting that the union of all finite sets related to the condition in set B is countable and thus Borel measurable with measure zero.

Areas of Agreement / Disagreement

Participants express differing views on the measurability of set B and the requirements for establishing it as measurable. There is no consensus on the approach to proving the measurability of set B, and uncertainty remains regarding the definitions and implications of the sets involved.

Contextual Notes

Participants reference the need for a Borel set and the concept of null sets in the context of Lebesgue measurability, indicating that the discussion may depend on specific definitions and assumptions related to these concepts.

Jamesandthegi
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I am trying to learn about Lebesgue measure. One of the questions I couldn't solve is this;

Show that the following sets are Lebesgue measurable and determine their measure

A = {x in [0,1) : the nth digit in the decimal expansion is equal to 7}

B = {x in [0,1) : all but finitely many digits in the decimal expansion are equal to 7}

Now, the book defines a set E to be Lebesgue measurable if E = A U B, where A is in the Borel $\sigma$-algebra and B is a null set (outer measure 0), but I don't see where that helps here. Any hints?
 
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Ok, I figured out A, but I'm not sure on B. Any help?
 
The measure is 0. The general idea is that for any n if n digits are 7, then the measure is (1/10)n. So let n -> ∞
 
Yes, but that doesn't prove that it's measurable. To prove its so, you have to write it as A U B, A borel and B null. Then the measure part I get.
 
If its null it is measurable. You can always union it with the empty set of you insist on having a Borel set to union with.
 
For a fixed finite set of decimal place not equal to seven the number of points is finite. So for instance the number of points if only the first 2 place are not 7 is 100. The number of finite subsets of a countable set is countable, I think. So the union of all of these finite sets is countable and thus Borel measurable zero.
 

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