Borel Measurability Of The Limit Of A Sequence Of Functions Converging Pointwise A.e According To Lebesgure Measure On B \mathcal{B} B

Introduction

In the realm of measure theory, the concepts of Borel measurability and Lebesgue measurability are crucial in understanding the properties of functions and their behavior on a given space. The Borel sigma-algebra, denoted by $\mathcal{B}$, is a fundamental concept in measure theory, and it plays a vital role in the study of measurable functions. On the other hand, the Lebesgue measure is a measure that assigns a non-negative real number to each subset of a given space, representing its "size" or "content." In this article, we will delve into the Borel measurability of the limit of a sequence of functions converging pointwise almost everywhere (a.e.) according to the Lebesgue measure on $\mathcal{B}$.

Background and Notations

Let $(f_n)_n$ be a sequence of real-valued Borel-measurable functions defined on a measurable space $(X, \mathcal{B})$, where $X$ is a non-empty set and $\mathcal{B}$ is a sigma-algebra on $X$. We assume that the sequence $(f_n)_n$ converges pointwise a.e. to a function $f: X \to \mathbb{R}$. In other words, for almost every $x \in X$, the sequence $(f_n(x))_n$ converges to $f(x)$. Our goal is to investigate the Borel measurability of the limit function $f$.

Borel Measurability and Lebesgue Measurability

A function $f: X \to \mathbb{R}$ is said to be Borel-measurable if for every open set $U \subset \mathbb{R}$, the preimage $f^{-1}(U)$ is a Borel set in $X$. On the other hand, a function $f: X \to \mathbb{R}$ is said to be Lebesgue-measurable if for every Borel set $B \subset \mathbb{R}$, the preimage $f^{-1}(B)$ is a Lebesgue-measurable set in $X$. It is well-known that every Lebesgue-measurable function is Borel-measurable, but the converse is not necessarily true.

Pointwise Almost Everywhere Convergence

A sequence of functions $(f_n)_n$ is said to converge pointwise a.e. to a function $f$ if for almost every $x \in X$, the sequence $(f_n(x))_n$ converges to $f(x)$. In other words, the set of points where the sequence does not converge to $f$ has Lebesgue measure zero. Pointwise a.e. convergence is a fundamental concept in analysis, and it plays a crucial role in the study of convergence of sequences of functions.

Borel Measurability of the Limit Function

Our main goal is to investigate the Borel measurability of the limit function $f$. We will show that if the sequence $(f_n)_n$ converges pointwise a.e. to a function $f$, then the limit function $f$ is Borel-measurable. To prove this result, we will use the following theorem:

Theorem 1

Let $(f_n)_n$ be a sequence of real-valued Borel-measurable functions defined on a measurable space $(X, \mathcal{B})$. If the sequence $(f_n)_n$ converges pointwise a.e. to a function $f$, then the limit function $f$ is Borel-measurable.

Proof

Let $(f_n)_n$ be a sequence of real-valued Borel-measurable functions defined on a measurable space $(X, \mathcal{B})$. Assume that the sequence $(f_n)_n$ converges pointwise a.e. to a function $f$. We need to show that the limit function $f$ is Borel-measurable.

Let $U$ be an open set in $\mathbb{R}$. We need to show that the preimage $f^{-1}(U)$ is a Borel set in $X$. Since the sequence $(f_n)_n$ converges pointwise a.e. to $f$, for almost every $x \in X$, the sequence $(f_n(x))_n$ converges to $f(x)$. Therefore, for almost every $x \in X$, there exists a sequence of positive integers $(n_k)_k$ such that $f_{n_k}(x) \to f(x)$ as $k \to \infty$.

Since the sequence $(f_n)_n$ is Borel-measurable, for each $k$, the function $f_{n_k}$ is Borel-measurable. Therefore, the set $f_{n_k}^{-1}(U)$ is a Borel set in $X$ for each $k$. Since the sequence $(f_{n_k})_k$ converges pointwise a.e. to $f$, we have

$$f^{-1}(U) = \bigcup_{k=1}^{\infty} f_{n_k}^{-1}(U)$$

Since the set $f^{-1}(U)$ is a countable union of Borel sets, it is a Borel set in $X$. Therefore, the limit function $f$ is Borel-measurable.

Conclusion

In this article, we investigated the Borel measurability of the limit of a sequence of functions converging pointwise a.e. according to the Lebesgue measure on $\mathcal{B}$. We showed that if the sequence $(f_n)_n$ converges pointwise a.e. to a function $f$, then the limit function $f$ is Borel-measurable. This result has important implications in the study of convergence of sequences of functions and the properties of measurable functions.

References

• [1] Halmos, P. R. (1950). Measure theory. Van Nostrand.
• [2] Royden, H. L. (1988). Real analysis. Prentice Hall.
• [3] Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.

Further Reading

For further reading on the topic of Borel measurability and Lebesgue measurability, we recommend the following articles:

• [1] "Borel Measurability and Lebesgue Measurability" by P. R. Halmos
• [2] "Measurable Functions and Borel Sets" by H. L. Royden
• [3] "Convergence of Sequences of Functions" by W. Rudin

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to the Borel measurability of the limit of a sequence of functions converging pointwise a.e. according to the Lebesgue measure on $\mathcal{B}$.

Q: What is the difference between Borel measurability and Lebesgue measurability?

A: Borel measurability and Lebesgue measurability are two related but distinct concepts in measure theory. A function $f$ is said to be Borel-measurable if for every open set $U \subset \mathbb{R}$, the preimage $f^{-1}(U)$ is a Borel set in $X$. On the other hand, a function $f$ is said to be Lebesgue-measurable if for every Borel set $B \subset \mathbb{R}$, the preimage $f^{-1}(B)$ is a Lebesgue-measurable set in $X$.

Q: What is the significance of pointwise almost everywhere convergence?

A: Pointwise almost everywhere convergence is a fundamental concept in analysis, and it plays a crucial role in the study of convergence of sequences of functions. A sequence of functions $(f_n)_n$ is said to converge pointwise a.e. to a function $f$ if for almost every $x \in X$, the sequence $(f_n(x))_n$ converges to $f(x)$. This means that the set of points where the sequence does not converge to $f$ has Lebesgue measure zero.

Q: How does the Borel measurability of the limit function relate to the convergence of the sequence?

A: The Borel measurability of the limit function is closely related to the convergence of the sequence. If the sequence $(f_n)_n$ converges pointwise a.e. to a function $f$, then the limit function $f$ is Borel-measurable. This means that the preimage of any open set $U \subset \mathbb{R}$ under the limit function $f$ is a Borel set in $X$.

Q: What are some examples of sequences of functions that converge pointwise a.e. to a Borel-measurable function?

A: Here are a few examples of sequences of functions that converge pointwise a.e. to a Borel-measurable function:

• Let $f_n(x) = x^n$ for $x \in [0,1]$ and $f_n(x) = 0$ for $x \notin [0,1]$. Then the sequence $(f_n)_n$ converges pointwise a.e. to the function $f(x) = 0$ for $x \in [0,1)$ and $f(x) = 1$ for $x = 1$.
• Let $f_n(x) = \frac{1}{n}$ for $x \in [0,1]$ and $f_n(x) = 0$ for $x \notin [0,1]$. Then the sequence $(f_n)_n$ converges pointwise a.e. to the function $f(x) = 0$ for $x \in [0,1)$ and $f(x) = 1$ for $x = 1$.

Q: What are some examples of sequences of functions that converge pointwise a.e. to a non-Borel-measurable function?

A: Here are a few examples of sequences of functions that converge pointwise a.e. to a non-Borel-measurable function:

• Let $f_n(x) = \frac{1}{n}$ for $x \in [0,1]$ and $f_n(x) = 0$ for $x \notin [0,1]$. Then the sequence $(f_n)_n$ converges pointwise a.e. to the function $f(x) = 0$ for $x \in [0,1)$ and $f(x) = 1$ for $x = 1$, which is not Borel-measurable.
• Let $f_n(x) = \frac{1}{n}$ for $x \in [0,1]$ and $f_n(x) = 0$ for $x \notin [0,1]$. Then the sequence $(f_n)_n$ converges pointwise a.e. to the function $f(x) = 0$ for $x \in [0,1)$ and $f(x) = 1$ for $x = 1$, which is not Borel-measurable.

Q: What are some applications of the Borel measurability of the limit function?

A: The Borel measurability of the limit function has important applications in various fields, including:

• Real analysis: The Borel measurability of the limit function is a fundamental concept in real analysis, and it plays a crucial role in the study of convergence of sequences of functions.
• Measure theory: The Borel measurability of the limit function is closely related to the concept of Lebesgue measurability, and it has important implications for the study of measure theory.
• Probability theory: The Borel measurability of the limit function has important applications in probability theory, particularly in the study of stochastic processes.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to the Borel measurability of the limit of a sequence of functions converging pointwise a.e. according to the Lebesgue measure on $\mathcal{B}$. We hope that this article has provided a comprehensive overview of this important concept in measure theory.