Proving Measurability of ##A## from ##E=A \cup B## with ##|B|=0##

In summary, the conversation discusses proving the measurability of a set A given that it is a subset of a measurable set E. It is mentioned that B is a null set and the speaker is seeking suggestions for constructing an elementary set H' such that the measure of its difference with A is less than a given epsilon.
  • #1
mahler1
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Homework Statement



Let ##E \subset \mathbb R^n## be a measurable set such that ##E=A \cup B## with ##|B|=0## (##B## is a null set). Show that ##A## is measurable.

The Attempt at a Solution



I know that given ##\epsilon##, there exists a ##\sigma##-elementary set ##H## such that ##E \subset H## and ##m_e(H-E)<\epsilon##. How can I construct a ##\sigma-##elementary set ##H'## such that ##m_e(H-A)<\epsilon##?. Any suggestions would be appreciated
 
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  • #2
What does |B| mean? Is it the measure of B or is the cardinality of B (in which case B is empty so that E= A).

In your last sentence do you not mean "How can I construct an elementary set H' such that [itex]m_e(H'- A)<\epsilon[/itex]"?
 
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  • #3
HallsofIvy said:
What does |B| mean? Is it the measure of B or is the cardinality of B (in which case B is empty so that E= A).

In your last sentence do you not mean "How can I construct an elementary set H' such that [itex]m_e(H'- A)<\epsilon[/itex]"?

Edited, thanks for the correction.
 

1. How can I prove the measurability of ##A## from ##E=A \cup B## with ##|B|=0##?

In order to prove the measurability of ##A## from ##E=A \cup B## with ##|B|=0##, you can use the fact that if ##|B|=0##, then ##B## is a null set. This means that the measure of ##B## is equal to 0, and therefore does not have any impact on the overall measure of ##E##. Thus, the measurability of ##E## is determined solely by the measurability of ##A##, and you can prove the measurability of ##A## by using the definition of measurability or any other suitable method.

2. What is the definition of measurability in the context of sets and measures?

In general, a set is considered measurable if it can be assigned a measure, which is a function that assigns a non-negative real number to each subset of the set. In the context of sets and measures, a set is said to be measurable if the measure of the set is well-defined and can be calculated using the properties and operations of the measure.

3. Why is it important to prove the measurability of a set?

Proving the measurability of a set is important because it allows us to accurately calculate and compare measures of different sets. In many areas of science, such as in physics and statistics, measurability is essential for making meaningful and reliable conclusions and predictions. Additionally, proving the measurability of a set can also help us understand the properties and behavior of the set and its elements.

4. Can a set be measurable if it has a measure of 0?

Yes, a set can still be considered measurable if it has a measure of 0. This is because a set with a measure of 0 can still be assigned a measure, and its measurability is determined by whether or not its measure is well-defined. In fact, sets with measures of 0 are often referred to as null sets and play an important role in measure theory.

5. Are there any other methods for proving the measurability of a set besides using the definition of measurability?

Yes, there are several other methods for proving the measurability of a set. These include using the properties and operations of measures, such as countable additivity and monotonicity, to show that the measure of the set is well-defined. Additionally, in some cases, the measurability of a set can be proven by showing that it is a subset of a larger, known measurable set.

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