# How Do You Prove the Equivalence of These Definitions of Measurability?

• AxiomOfChoice
In summary, in real analysis, measurability is defined as a set E being (Lebesgue) measurable if for every \epsilon > 0 there exists an open set \mathcal O \supseteq E such that m_*(\mathcal O \setminus E) < \epsilon. Another equivalent definition is that for every \epsilon > 0 there exists a closed set F \subseteq E such that m_*(E\setminus F) < \epsilon. The equivalence of these definitions is a problem in Stein-Shakarchi's textbook and a practice problem for the final exam. One possible approach to showing the equivalence is using the open/closed duality and choosing F to be the complement of the open
AxiomOfChoice
One possible definition of measurability is this: A set $$E \subseteq \mathbb R^d$$ is (Lebesgue) measurable if for every $$\epsilon > 0$$ there exists an open set $$\mathcal O \supseteq E$$ such that $$m_*(\mathcal O \setminus E) < \epsilon$$. Here, $$m_*$$ indicates Lebesgue outer measure.

Apparently, an equivalent definition is this: "For every $$\epsilon > 0$$ there exists a closed set $$F \subseteq E$$ such that $$m_*(E\setminus F) < \epsilon$$."

Showing the equivalence of these definitions was a practice problem recently for the final exam in my real analysis class. But I couldn't get it, and even though I'm on break now, it's bugging me. Can someone help? Thanks! (This is also apparently a problem in Stein-Shakarchi's textbook, Real Analysis.)

==>: Suppose E is measurable. Then E^c is measurable. Let O be the open set associated to E^c as in the definition of measurability. Then use F=O^c.

<==: Same thing, just use the open/closed duality in the same way.

## 1. What is measure theory?

Measure theory is a branch of mathematics that deals with the concept of "measure" in order to provide a mathematical framework for understanding and analyzing the properties of sets and functions. It is particularly useful in studying and analyzing the properties of real-valued functions and their integrals.

## 2. What is a measure?

A measure is a mathematical concept that assigns a numerical value to subsets of a given set, which represents the "size" or "extent" of the subset. In measure theory, measures are used to quantify the properties of sets and functions.

## 3. What are the key components of measure theory?

The key components of measure theory include measure spaces, measurable functions, and integration. A measure space consists of a set, a sigma-algebra (a collection of subsets of the set), and a measure function. Measurable functions are those that preserve the "size" or "extent" of sets, and integration is a way to calculate the size of a set or function.

## 4. How is measure theory applied in real-world problems?

Measure theory has many practical applications, particularly in fields such as physics, economics, and engineering. It is used to analyze and model complex systems, measure probabilities and uncertainties, and solve optimization problems.

## 5. What are some common challenges in understanding measure theory?

Some common challenges in understanding measure theory include its abstract nature, the use of formal mathematical notation and terminology, and the need for a solid foundation in real analysis. It can also be challenging to apply measure theory to real-world problems, as it often requires advanced mathematical techniques and critical thinking skills.

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