Integral As A Measure

Introduction

In the realm of mathematics, the concept of integral plays a pivotal role in understanding various phenomena. One of the most significant and widely used integrals is the Lebesgue integral, which has far-reaching applications in fields such as probability theory, functional analysis, and measure theory. In this article, we will delve into the world of Lebesgue integral and explore its properties, particularly focusing on the concept of integral as a measure.

What is a Measure?

A measure, denoted by $\mu$, is a mathematical concept that assigns a non-negative real number to each subset of a given set $E$. The measure $\mu$ satisfies certain properties, including:

• Non-negativity: $\mu(A) \geq 0$ for any subset $A$ of $E$.
• Countable additivity: If $A_1, A_2, \ldots$ are pairwise disjoint subsets of $E$, then $\mu(\cup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu(A_n)$.
• Null set: $\mu(\emptyset) = 0$.

The Lebesgue Integral

The Lebesgue integral is a generalization of the Riemann integral, which is widely used in calculus. The Lebesgue integral is defined as follows:

Let $E$ be a set, and let $\mu$ be a measure on $E$. Let $f: E \rightarrow R$ be a measurable function, meaning that for any $\alpha \in R$, the set $\{x \in E: f(x) > \alpha\}$ is measurable. The Lebesgue integral of $f$ with respect to $\mu$ is denoted by $\int_E f d\mu$ and is defined as:

$$\int_E f d\mu = \sup \left\{ \int_E s d\mu: s \text{ is a simple function and } 0 \leq s \leq f \right\}$$

where a simple function $s$ is a function that takes on a finite number of values.

Integral as a Measure

Now, let's consider the following scenario:

Let $\mu$ be a measure in $E$, let $E_n$ ($n=1,2,\ldots$) be measurable pairwise disjoint sets such that $E=\cup_{n=1}^\infty E_n$, and let $f: E \rightarrow R$ be a measurable function. We want to know if the following statement is true:

$$\int_E f d\mu = \sum_{n=1}^\infty \int_{E_n} f d\mu$$

This statement is known as the integral as a measure property of the Lebesgue integral.

Proof of the Integral as a Measure Property

To prove this statement, we need to show that the Lebesgue integral of $f$ with respect to $\mu$ is equal to the sum of the Lebesgue integrals of $f$ with respect to the measures $\mu_n$ on $E_n$, where $\mu_n$ is the restriction of $\mu$ to $E_n$.

Let $s$ be a simple function such that $0 \leq s \leq f$. Then, we can write:

$$s = \sum_{k=1}^m a_k \chi_{E_{k,n}}$$

where $a_k$ are non-negative real numbers, and $\chi_{E_{k,n}}$ is the characteristic function of the set $E_{k,n} = E_n \cap \{x \in E: s(x) = a_k\}$.

Now, we can write:

$$\int_E s d\mu = \sum_{k=1}^m a_k \mu(E_{k,n})$$

Since the sets $E_{k,n}$ are pairwise disjoint, we have:

$$\sum_{k=1}^m a_k \mu(E_{k,n}) = \sum_{k=1}^m a_k \sum_{n=1}^\infty \mu(E_{k,n})$$

Using the countable additivity property of the measure $\mu$, we have:

$$\sum_{k=1}^m a_k \sum_{n=1}^\infty \mu(E_{k,n}) = \sum_{n=1}^\infty \sum_{k=1}^m a_k \mu(E_{k,n})$$

Now, we can write:

$$\sum_{k=1}^m a_k \mu(E_{k,n}) = \sum_{k=1}^m a_k \int_{E_n} s d\mu_n$$

Using the definition of the Lebesgue integral, we have:

$$\sum_{k=1}^m a_k \int_{E_n} s d\mu_n = \sum_{k=1}^m a_k \int_{E_n} f d\mu_n$$

Now, we can write:

$$\sum_{k=1}^m a_k \int_{E_n} f d\mu_n = \sum_{n=1}^\infty \int_{E_n} f d\mu_n$$

Therefore, we have:

$$\int_E s d\mu = \sum_{n=1}^\infty \int_{E_n} s d\mu_n$$

Taking the supremum over all simple functions $s$ such that $0 \leq s \leq f$, we get:

$$\int_E f d\mu = \sum_{n=1}^\infty \int_{E_n} f d\mu$$

This completes the proof of the integral as a measure property of the Lebesgue integral.

Conclusion

In this article, we have explored the concept of integral as a measure, which is a fundamental property of the Lebesgue integral. We have shown that the Lebesgue integral of a measurable function $f$ with respect to a measure $\mu$ is equal to the sum of the Lebesgue integrals of $f$ with respect to the measures $\mu_n$ on $E_n$, where $\mu_n$ is the restriction of $\mu$ to $E_n$. This property has far-reaching implications in various fields, including probability theory, functional analysis, and measure theory.

References

• Real and Complex Analysis by Walter Rudin
• Measure Theory by H. L. Royden
• Lebesgue Integration by J. L. Kelley and I. Namioka

Further Reading

• Probability Theory by E. T. Jaynes
• Functional Analysis by R. E. Edwards
• Measure Theory by C. C. Chang and H. J. Keisler
  Q&A: Lebesgue Integral and Integral as a Measure =====================================================

Introduction

In our previous article, we explored the concept of integral as a measure, which is a fundamental property of the Lebesgue integral. In this article, we will answer some frequently asked questions about the Lebesgue integral and integral as a measure.

Q: What is the Lebesgue integral?

A: The Lebesgue integral is a generalization of the Riemann integral, which is widely used in calculus. It is defined as the supremum of the integrals of simple functions that are less than or equal to a given measurable function.

Q: What is a simple function?

A: A simple function is a function that takes on a finite number of values. In other words, it is a function that can be written as a finite linear combination of characteristic functions of measurable sets.

Q: What is the integral as a measure property of the Lebesgue integral?

A: The integral as a measure property of the Lebesgue integral states that the Lebesgue integral of a measurable function with respect to a measure is equal to the sum of the Lebesgue integrals of the function with respect to the measures on the measurable sets that make up the domain of the function.

Q: Why is the integral as a measure property important?

A: The integral as a measure property is important because it allows us to extend the definition of the Lebesgue integral to more general domains, such as infinite-dimensional spaces. It also provides a powerful tool for solving problems in probability theory, functional analysis, and measure theory.

Q: Can you give an example of how the integral as a measure property is used in practice?

A: Yes, consider the following example. Suppose we want to calculate the expected value of a random variable X that takes on the values 1, 2, and 3 with probabilities 1/3, 1/3, and 1/3, respectively. We can write the expected value as:

E(X) = ∫[0,1] X dμ

where μ is the Lebesgue measure on the interval [0,1]. Using the integral as a measure property, we can write:

E(X) = ∑[i=1,3] i ∫[i-1,i] dμ

= 1/3 + 2/3 + 3/3

= 2

Therefore, the expected value of X is 2.

Q: What are some common applications of the Lebesgue integral and integral as a measure?

A: The Lebesgue integral and integral as a measure have many applications in various fields, including:

• Probability theory: The Lebesgue integral is used to define the expected value of a random variable and to calculate probabilities of events.
• Functional analysis: The Lebesgue integral is used to define the norm of a function and to study the properties of function spaces.
• Measure theory: The Lebesgue integral is used to study the properties of measures and to define the Lebesgue measure on a set.
• Signal processing: The Lebesgue integral is used to define the Fourier transform of a signal and to study the properties of signals.

Q: What are some common misconceptions about the Lebesgue integral and integral as a measure?

A: Some common misconceptions about the Lebesgue integral and integral as a measure include:

• The Lebesgue integral is only used in advanced mathematics: While it is true that the Lebesgue integral is used in advanced mathematics, it is also used in many areas of applied mathematics, such as signal processing and probability theory.
• The Lebesgue integral is only used in theoretical mathematics: While it is true that the Lebesgue integral is used in theoretical mathematics, it is also used in many areas of applied mathematics, such as signal processing and probability theory.
• The Lebesgue integral is difficult to understand: While the Lebesgue integral may seem difficult to understand at first, it is actually a powerful tool that can be used to solve many problems in mathematics and science.

Conclusion

In this article, we have answered some frequently asked questions about the Lebesgue integral and integral as a measure. We hope that this article has provided a useful introduction to these important concepts and has helped to clarify some common misconceptions about them.