Proving $\lim_{n\to \infty}f(x+n)=0$ For $f \in L^1(\mathbb{R})$

Introduction

In the realm of real analysis and measure theory, a fundamental concept is the behavior of functions within the space of Lebesgue integrable functions, denoted as $L^1(\mathbb{R})$. Specifically, we delve into a crucial property: for a function $f$ belonging to $L^1(\mathbb{R})$, we aim to demonstrate that the limit of $f(x+n)$ as $n$ approaches infinity equals zero for almost every $x$ in the real numbers, where $n$ is a natural number. This article provides a detailed exploration of this concept, offering a comprehensive proof and shedding light on its implications within the broader context of Lebesgue integration and measure theory. We will navigate through the intricacies of Lebesgue measure, the properties of $L^1$ functions, and the significance of “almost everywhere” convergence, ensuring a robust understanding of the subject matter. Understanding this limit is pivotal in various applications, including signal processing, probability theory, and the study of dynamical systems. By establishing this result, we gain deeper insights into the long-term behavior of integrable functions under translation, a cornerstone in functional analysis and harmonic analysis. This exploration is not only theoretically enriching but also practically relevant, providing tools to analyze functions and their transformations in a rigorous and meaningful way. The proof involves leveraging the properties of the Lebesgue integral, emphasizing the absolute integrability of $f$, and employing techniques from real analysis to establish the pointwise convergence to zero almost everywhere.

Preliminaries: Lebesgue Integrable Functions and Measure Theory

Before diving into the proof, it's crucial to lay a solid foundation by revisiting key definitions and concepts from Lebesgue integration and measure theory. This section provides a comprehensive overview, ensuring that the subsequent arguments are clear and well-understood. We begin by defining Lebesgue measure, a cornerstone of real analysis that generalizes the concept of length, area, and volume to a broader class of sets than the familiar intervals, rectangles, and cubes. Lebesgue measure allows us to assign a size to intricate sets, including those that are not easily described using traditional geometric methods. It is a complete measure, meaning that every subset of a set of measure zero is also measurable and has measure zero. This property is vital for dealing with functions that may be undefined or behave erratically on small sets.

Next, we introduce the space of Lebesgue integrable functions, denoted as $L^1(\mathbb{R})$. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be Lebesgue integrable if it is measurable and the integral of its absolute value over the entire real line is finite. Formally, this is expressed as $\int_{\mathbb{R}} |f| dm < \infty$, where $m$ represents the Lebesgue measure. The absolute integrability is a crucial condition, ensuring that the integral is well-defined and that the function's “total area” under the curve is finite. This class of functions is central to many areas of mathematics, including Fourier analysis, probability theory, and partial differential equations. The $L^1$ space is a Banach space, meaning it is a complete normed vector space, which provides a robust framework for studying convergence and approximation properties of functions. Functions in $L^1(\mathbb{R})$ can exhibit complex behaviors, but their integrability ensures a certain degree of control and predictability, making them amenable to analysis using powerful tools from functional analysis.

We also emphasize the notion of “almost everywhere” (a.e.) convergence. A property is said to hold almost everywhere if it holds for all points except for a set of Lebesgue measure zero. This concept is essential because it allows us to disregard sets that are negligible in the context of integration. For instance, if two functions are equal almost everywhere, their Lebesgue integrals are the same. This perspective is particularly useful when dealing with functions that may have singularities or discontinuities on small sets. In the context of this problem, showing that $\lim_{n\to \infty} f(x+n) = 0$ almost everywhere means that the limit holds for all $x$ except for a set of measure zero. This is a weaker form of pointwise convergence but is often sufficient in applications involving integration and measure theory. Understanding the nuances of almost everywhere convergence is critical for interpreting results involving Lebesgue integrable functions and for appreciating the power and flexibility of Lebesgue integration.

Problem Statement and Strategy

Our primary objective is to prove that for any function $f \in L^1(\mathbb{R})$, the limit $\lim_{n\to \infty} f(x+n) = 0$ holds for almost every $x \in \mathbb{R}$, where $n$ is a natural number. This statement suggests that as we shift the function $f$ further and further to the left along the real line (by adding increasingly large natural numbers to $x$), the function values tend to zero for almost all points. The condition $f \in L^1(\mathbb{R})$ plays a crucial role, indicating that the function is absolutely integrable, which means that the integral of its absolute value over the real line is finite. This integrability condition is essential for controlling the function's behavior and ensuring that the limit result holds.

To tackle this problem, we will employ a strategy that combines the properties of Lebesgue integrable functions with the concept of measure preservation under translation. The general approach involves showing that the integral of $|f(x+n)|$ over any finite interval tends to zero as $n$ approaches infinity. This is a key step because it connects the global integrability condition of $f$ with the local behavior of its translates. We leverage the fact that the Lebesgue measure is translation-invariant, which means that shifting a measurable set does not change its measure. This property is fundamental in understanding how translations affect integrals and limits. By considering the integral of $|f(x+n)|$ over a finite interval, we can establish a connection between the function's behavior and its integral, allowing us to use the integrability condition effectively.

Specifically, we will first consider the integral of $|f(x+n)|$ over an arbitrary finite interval $[a, b]$. We will demonstrate that this integral converges to zero as $n$ goes to infinity. This step is crucial because it implies that the “mass” of the function $f$ is being pushed away from any finite interval as $n$ increases. We then use this result to show that the set of points where the limit $\lim_{n\to \infty} |f(x+n)|$ is not zero must have measure zero. This involves using the Borel-Cantelli lemma, a powerful tool in measure theory that helps us analyze the long-term behavior of sequences of events. The Borel-Cantelli lemma allows us to deduce that certain events occur only finitely often with probability one, which translates into measure zero in our context. By combining the integral convergence result with the Borel-Cantelli lemma, we can rigorously establish that $\lim_{n\to \infty} f(x+n) = 0$ for almost every $x \in \mathbb{R}$. This strategy provides a clear pathway to the solution, highlighting the interplay between integrability, translation invariance, and measure-theoretic tools.

Proof: $\lim_{n\to \infty}f(x+n)=0$ for Almost Every $x \in \mathbb{R}$

Let $f \in L^1(\mathbb{R})$. Our goal is to demonstrate that $\lim_{n\to \infty} f(x+n) = 0$ for almost every $x \in \mathbb{R}$.

1. Consider the Integral of the Translated Function: We begin by examining the integral of the absolute value of the translated function $|f(x+n)|$ over the real line. Since $f \in L^1(\mathbb{R})$, we know that $\int_{\mathbb{R}} |f(x)| dm(x) < \infty$. We aim to show that the integral of $|f(x+n)|$ over a finite interval approaches zero as $n$ tends to infinity. Due to the translation invariance of the Lebesgue measure, we have:

   $\int_{\mathbb{R}} |f(x+n)| dm(x) = \int_{\mathbb{R}} |f(x)| dm(x) < \infty$.

   This equality holds because shifting the function $f$ by $n$ units does not change the value of its integral over the entire real line. The translation invariance of the Lebesgue measure ensures that the “total area” under the curve of $|f(x+n)|$ remains the same as that of $|f(x)|$. This is a crucial observation, as it allows us to relate the integrability of $f$ to the behavior of its translates.

2. Integral over a Finite Interval: Now, let's consider an arbitrary finite interval $[a, b]$. We want to show that $\int_{a}^{b} |f(x+n)| dx$ approaches zero as $n$ goes to infinity. To do this, we can rewrite the integral using a change of variables. Let $y = x + n$, so $x = y - n$, and $dx = dy$. The interval $[a, b]$ transforms to $[a+n, b+n]$. Thus, we have:

   $\int_{a}^{b} |f(x+n)| dx = \int_{a+n}^{b+n} |f(y)| dy$.

   As $n$ approaches infinity, the interval $[a+n, b+n]$ shifts further and further to the right along the real line. The integral $\int_{a+n}^{b+n} |f(y)| dy$ represents the “area” under the curve of $|f(y)|$ within this shifted interval. Since $f \in L^1(\mathbb{R})$, the total area under the curve of $|f(y)|$ is finite. As we consider intervals $[a+n, b+n]$ that move further away from the origin, the “area” under the curve within these intervals must approach zero. This is because if the integral over these intervals did not approach zero, the total integral over the entire real line would be infinite, contradicting the fact that $f \in L^1(\mathbb{R})$. Formally, we can express this as:

   $\lim_{n \to \infty} \int_{a+n}^{b+n} |f(y)| dy = 0$.

   This convergence to zero is a direct consequence of the absolute integrability of $f$. It highlights the connection between the global integrability condition and the local behavior of the function under translation.

3. Defining the Set Where the Limit Fails: Let's define a set $E_{\epsilon}$ for $\epsilon > 0$ as the set of points $x$ where $\limsup_{n \to \infty} |f(x+n)| > \epsilon$. In other words, $E_{\epsilon}$ contains all points $x$ such that the sequence $|f(x+n)|$ does not converge to zero for a given $\epsilon$. Our goal is to show that the Lebesgue measure of $E_{\epsilon}$ is zero for any $\epsilon > 0$.

   To do this, we consider the sets $E_{\epsilon, k} = \{x \in \mathbb{R} : |f(x+n)| > \epsilon \text{ for infinitely many } n \geq k\}$, where $k$ is a natural number. The set $E_{\epsilon, k}$ represents the points $x$ for which $|f(x+n)|$ exceeds $\epsilon$ infinitely often, starting from $n = k$. The relationship between $E_{\epsilon}$ and $E_{\epsilon, k}$ is that $E_{\epsilon} = \bigcup_{k=1}^{\infty} E_{\epsilon, k}$. This means that a point $x$ belongs to $E_{\epsilon}$ if and only if it belongs to at least one of the sets $E_{\epsilon, k}$.

   Now, we define sets $A_{n} = \{x \in [0,1] : |f(x+n)| > \epsilon\}$ for each natural number $n$. The set $A_{n}$ represents the points in the interval $[0, 1]$ where $|f(x+n)|$ exceeds $\epsilon$. We want to analyze the measure of these sets and their intersections.

4. Applying the Borel-Cantelli Lemma: To proceed, we need to show that the sum of the measures of the sets $A_{n}$ is finite. This is a crucial step in applying the Borel-Cantelli lemma. We have:

   $\sum_{n=1}^{\infty} m(A_{n}) = \sum_{n=1}^{\infty} m(\{x \in [0,1] : |f(x+n)| > \epsilon\})$.

   Since the Lebesgue measure is translation-invariant, we can rewrite the measure of $A_{n}$ in terms of the integral of the indicator function of the set where $|f(x+n)| > \epsilon$:

   $m(A_{n}) = \int_{0}^{1} \mathbb{1}_{|f(x+n)| > \epsilon} dx$.

   Now, we can relate this to the integral of $|f(x+n)|$ over the interval $[0, 1]$. If $|f(x+n)| > \epsilon$, then $\epsilon \mathbb{1}_{|f(x+n)| > \epsilon} \leq |f(x+n)|$. Therefore:

   $\epsilon m(A_{n}) = \epsilon \int_{0}^{1} \mathbb{1}_{|f(x+n)| > \epsilon} dx \leq \int_{0}^{1} |f(x+n)| dx$.

   Summing over $n$, we get:

   $\epsilon \sum_{n=1}^{\infty} m(A_{n}) \leq \sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx$.

   Now, we can interchange the summation and integration (by the monotone convergence theorem or the dominated convergence theorem, since the terms are non-negative):

   $\sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx = \int_{0}^{1} \sum_{n=1}^{\infty} |f(x+n)| dx$.

   However, we need to show that this sum is finite. To do this, we consider the integral of $|f|$ over the real line. Since $f \in L^1(\mathbb{R})$, we have:

   $\int_{\mathbb{R}} |f(x)| dx < \infty$.

   We can express the integral over the real line as a sum of integrals over intervals of length 1:

   $\int_{\mathbb{R}} |f(x)| dx = \sum_{n=-\infty}^{\infty} \int_{n}^{n+1} |f(x)| dx < \infty$.

   This implies that the sum of the integrals over these intervals is finite. However, we are interested in the sum $\sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx$. To relate this to the integral of $|f|$, we can use the translation invariance of the Lebesgue integral. We have:

   $\int_{0}^{1} |f(x+n)| dx = \int_{n}^{n+1} |f(y)| dy$,

   where $y = x + n$. Thus, the sum becomes:

   $\sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx = \sum_{n=1}^{\infty} \int_{n}^{n+1} |f(y)| dy$.

   However, this sum is not necessarily finite, as it only includes the integrals over intervals $[n, n+1]$ for positive $n$. To overcome this issue, we need to use a different approach.

5. Revised Approach Using the Borel-Cantelli Lemma: Let's reconsider the sets $A_{n} = \{x \in [0,1] : |f(x+n)| > \epsilon\}$. We want to show that the sum of the measures of these sets is finite. We have:

   $m(A_{n}) = \int_{0}^{1} \mathbb{1}_{|f(x+n)| > \epsilon} dx \leq \frac{1}{\epsilon} \int_{0}^{1} |f(x+n)| dx$.

   Summing over $n$, we get:

   $\sum_{n=1}^{\infty} m(A_{n}) \leq \frac{1}{\epsilon} \sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx$.

   Now, we interchange the summation and integration:

   $\sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx = \int_{0}^{1} \sum_{n=1}^{\infty} |f(x+n)| dx$.

   This step requires justification. We can't directly apply the monotone convergence theorem here because the sum inside the integral might not be monotone. Instead, we can use a different approach. We know that $\int_{\mathbb{R}} |f(x)| dx < \infty$. Let's consider the function $g(x) = \sum_{n=1}^{\infty} |f(x+n)|$. We want to show that $\int_{0}^{1} g(x) dx < \infty$.

   We can rewrite the integral of $g$ over $[0, 1]$ as:

   $\int_{0}^{1} g(x) dx = \int_{0}^{1} \sum_{n=1}^{\infty} |f(x+n)| dx = \sum_{n=1}^{\infty} \int_{0}^{1} |f(x+n)| dx$.

   Using the translation invariance of the Lebesgue integral, we have:

   $\int_{0}^{1} |f(x+n)| dx = \int_{n}^{n+1} |f(y)| dy$,

   where $y = x + n$. Thus, the sum becomes:

   $\sum_{n=1}^{\infty} \int_{n}^{n+1} |f(y)| dy \leq \int_{1}^{\infty} |f(y)| dy$.

   Since $\int_{\mathbb{R}} |f(x)| dx < \infty$, we know that $\int_{1}^{\infty} |f(y)| dy < \infty$. This implies that $\sum_{n=1}^{\infty} m(A_{n}) < \infty$.

   Now, we can apply the Borel-Cantelli lemma. The Borel-Cantelli lemma states that if $\sum_{n=1}^{\infty} m(A_{n}) < \infty$, then $m(\limsup_{n \to \infty} A_{n}) = 0$, where $\limsup_{n \to \infty} A_{n}$ is the set of points that belong to infinitely many of the $A_{n}$. In our case, $\limsup_{n \to \infty} A_{n}$ is the set of points $x \in [0, 1]$ for which $|f(x+n)| > \epsilon$ for infinitely many $n$. Thus, we have:

   $m(\limsup_{n \to \infty} A_{n}) = 0$.

   This means that the set of points $x \in [0, 1]$ for which $|f(x+n)| > \epsilon$ for infinitely many $n$ has measure zero. Let $B_{\epsilon} = \{x \in [0, 1] : \limsup_{n \to \infty} |f(x+n)| > \epsilon\}$. Then $m(B_{\epsilon}) = 0$.

6. Extending to the Real Line: We have shown that for any $\epsilon > 0$, the set of points in $[0, 1]$ where $\limsup_{n \to \infty} |f(x+n)| > \epsilon$ has measure zero. Now, we need to extend this result to the entire real line. We can partition the real line into intervals of length 1: $\mathbb{R} = \bigcup_{k=-\infty}^{\infty} [k, k+1]$. Let $B_{\epsilon, k} = \{x \in [k, k+1] : \limsup_{n \to \infty} |f(x+n)| > \epsilon\}$. Using a similar argument as above, we can show that $m(B_{\epsilon, k}) = 0$ for each integer $k$.

   Now, let $E_{\epsilon} = \{x \in \mathbb{R} : \limsup_{n \to \infty} |f(x+n)| > \epsilon\}$. Then $E_{\epsilon} = \bigcup_{k=-\infty}^{\infty} B_{\epsilon, k}$. Since the measure of a countable union of sets of measure zero is also zero, we have:

   $m(E_{\epsilon}) = m(\bigcup_{k=-\infty}^{\infty} B_{\epsilon, k}) \leq \sum_{k=-\infty}^{\infty} m(B_{\epsilon, k}) = 0$.

   This shows that for any $\epsilon > 0$, the set of points where $\limsup_{n \to \infty} |f(x+n)| > \epsilon$ has measure zero.

7. Final Step: Almost Everywhere Convergence: Finally, let $E = \{x \in \mathbb{R} : \lim_{n \to \infty} f(x+n) \neq 0\}$. We can express $E$ as a countable union of sets:

   $E = \bigcup_{j=1}^{\infty} \{x \in \mathbb{R} : \limsup_{n \to \infty} |f(x+n)| > \frac{1}{j}\} = \bigcup_{j=1}^{\infty} E_{\frac{1}{j}}$.

   Since each $E_{\frac{1}{j}}$ has measure zero, the measure of their countable union is also zero:

   $m(E) = m(\bigcup_{j=1}^{\infty} E_{\frac{1}{j}}) \leq \sum_{j=1}^{\infty} m(E_{\frac{1}{j}}) = 0$.

   Therefore, the set of points where $\lim_{n \to \infty} f(x+n) \neq 0$ has measure zero. This implies that $\lim_{n \to \infty} f(x+n) = 0$ for almost every $x \in \mathbb{R}$.

Conclusion

In conclusion, we have rigorously demonstrated that for any function $f$ belonging to the space $L^1(\mathbb{R})$, the limit of $f(x+n)$ as $n$ approaches infinity is zero for almost every $x$ in the real numbers. This result underscores a fundamental property of Lebesgue integrable functions, highlighting their behavior under translation and their tendency to “vanish” at infinity in a measure-theoretic sense. The proof involved a careful interplay between the properties of Lebesgue integration, the translation invariance of Lebesgue measure, and the application of the Borel-Cantelli lemma. We first showed that the integral of the translated function $|f(x+n)|$ over any finite interval converges to zero as $n$ goes to infinity, a direct consequence of the absolute integrability of $f$. This established a crucial link between the global integrability condition and the local behavior of the function under translation. We then defined the set of points where the limit $\lim_{n \to \infty} |f(x+n)|$ does not equal zero and utilized the Borel-Cantelli lemma to prove that this set has measure zero. The Borel-Cantelli lemma provided a powerful tool for analyzing the long-term behavior of sequences of events, allowing us to deduce that certain events occur only finitely often, which translates to measure zero in our context. Finally, we extended the result from a finite interval to the entire real line by considering a countable union of intervals and applying the properties of Lebesgue measure. The significance of this result extends beyond the theoretical realm of real analysis and measure theory. It has practical implications in various fields, including signal processing, where understanding the long-term behavior of signals is crucial, and probability theory, where the convergence of random variables is a central theme. The property also finds applications in the study of dynamical systems, where the behavior of functions under iteration is of interest. By establishing that $\lim_{n \to \infty} f(x+n) = 0$ almost everywhere for $f \in L^1(\mathbb{R})$, we gain a deeper understanding of the nature of integrable functions and their transformations. This result not only enhances our theoretical knowledge but also provides valuable tools for analyzing functions and their applications in diverse areas of mathematics and science. The techniques used in this proof, such as leveraging the translation invariance of the Lebesgue measure and applying the Borel-Cantelli lemma, are fundamental in measure theory and serve as essential tools in the broader field of mathematical analysis. This exploration highlights the beauty and power of mathematical reasoning, showcasing how abstract concepts can lead to concrete and meaningful results.