Question

(The Vitali Convergence Theorem) Suppose \(1 \leq p<\infty\) and \(\left\\{f_{n}\right\\}_{1}^{\infty} \subset L^{p}\). In order for \(\left\\{f_{n}\right\\}\) to be Cauchy in the \(L^{p}\) norm it is necessary and sufficient for the following three conditions to hold: (i) \(\left\\{f_{n}\right\\}\) is Cauchy in measure; (ii) the sequence \(\left\\{\left|f_{n}\right|^{P}\right\\}\) is uniformly integrable (see Exercise 11 in \(\S 3.2\) ); and (iii) for every \(\in>0\) there exists \(E \subset X\) such that \(\mu(E)<\infty\) and \(\int_{E^{c}}\left|f_{n}\right|^{p}<\epsilon\) for all \(n\). (To prove the sufficiency: Given \(\epsilon>0\), let \(E\) be as in (iii), and let \(A_{\operatorname{mn}}=\\{x \in E\) : \(\left.\left|f_{m}(x)-f_{n}(x)\right| \geq \epsilon\right\\}\). Then the integrals of \(\left|f_{n}-f_{m}\right|^{p}\) over \(E \backslash A_{m n}, A_{m n}\), and \(E^{c}\) are small when \(m\) and \(n\) are large - for three different reasons.)

Short answer

The sequence \(\{f_n\}\) is Cauchy in the \(L^p\) norm if it satisfies being Cauchy in measure, uniformly integrable, and having a subset \(E\) with properties involving small integrals outside \(E\).

Step-by-step solution

Understand the Conditions

The problem involves establishing the Vitali Convergence Theorem, which requires proving that if the sequence \(\{f_n\}\) is Cauchy in the \(L^p\) norm, it meets three specific conditions: being Cauchy in measure, uniformly integrable, and for any \(\epsilon > 0\), there is a subset \(E\) such that \(\mu(E) < \infty\) and \(\int_{E^c} |f_n|^p < \epsilon\) for all \(n\).

Show Cauchy in Measure

Given \(\epsilon > 0\), since \(\{f_n\}\) is Cauchy in measure, for every \(\delta > 0\), there exists \(N\) such that for all \(m, n > N\), the measure \(\mu(\{x : |f_n(x) - f_m(x)| \, \ge \epsilon\}) < \delta\). This establishes the Cauchy property in measure.

Ensure Uniform Integrability

Since the sequence \(\{|f_n|^p\}\) is uniformly integrable, for any \(\epsilon > 0\), there exists \(\delta > 0\) such that for any measurable set \(A\) with \(\mu(A) < \delta\), \(\int_A |f_n|^p < \epsilon\) for all \(n\). This ensures that the integrals do not exceed \(\epsilon\) for small sets.

Use the Subset Condition

For any \(\epsilon > 0\), there exists a subset \(E\) with finite measure ensuring that \(\int_{E^c} |f_n|^p < \epsilon\) for all \(n\). This is important because it bounds the measure of the 'bad' part of the function outside \(E\).

Consider Integrals Over Different Sets

Decompose the integral of \(|f_m - f_n|^p\) into three parts: over \(E \setminus A_{mn}\), \(A_{mn}\), and \(E^c\). For each part, demonstrate that the integral is small when \(m\) and \(n\) are large. Use that \(\{f_n\}\) is Cauchy in measure and uniformly integrable to show smallness over \(E \setminus A_{mn}\). Utilize the set \(E^c\) condition for the third integral. Finally, show that the measure of \(A_{mn}\) is small by definition of Cauchy in measure, making the integral over it small.

Conclude the Proof

Combine all parts to confirm that for large \(m, n\), the \(L^p\) norm of the difference between \(f_m\) and \(f_n\) is sufficiently small, thereby proving the sufficiency of the conditions for \(\{f_n\}\) to be Cauchy in the \(L^p\) norm.

Key concepts

Cauchy sequence

A **Cauchy sequence** is a vital concept in the field of mathematics, especially in analysis, where it describes a sequence whose elements become arbitrarily close to each other as the sequence progresses. The formal definition can be expressed as: for any small number, say \( \epsilon > 0 \), there exists an integer \( N \) such that for all integers \( m, n > N \), the difference between the terms of the sequence \( |a_m - a_n| \) is less than \( \epsilon \).
This implies that the sequence converges to a point, even if that point isn't known ahead of time.
In the context of the Vitali Convergence Theorem, the sequence \( \{ f_n \} \) being Cauchy in the \( L^p \) norm means that the \( L^p \) norms of the differences \( ||f_m - f_n||_p \) become arbitrarily small for large \( m \) and \( n \). This property is fundamental for sequences where convergence is not visibly apparent, as it assures that the sequence is converging as intended, even in the complex space of \( L^p \) functions.

L^p norm

The **\( L^p \) norm** is a way to measure the size of functions in the space \( L^p \). It generalizes the concept of length for functions and provides a way to quantify their magnitude.
For a given function \( f \) defined on a measure space \( X \), the \( L^p \) norm is calculated as \( ||f||_p = \left( \int_X |f(x)|^p \, dx \right)^{1/p} \).
This formula is interpreted in the following way:

• \( |f(x)|^p \) denotes the function's absolute value raised to the power of \( p \).
• The integral \( \int_X \) sums this value over the entire domain \( X \).
• Finally, taking the \( p \)-th root of the result gives the \( L^p \) norm.

This norm is crucial in functional analysis and probability theory because it encapsulates both the "shape" and "size" of \( f \), rendering it indispensable in discussions about convergence like in the Vitali Convergence Theorem.

Uniform integrability

**Uniform integrability** is a condition that helps in understanding convergence of sequences of functions, ensuring that the members of the sequence do not "escape" to infinity.
To say that a sequence \( \{f_n \} \) is uniformly integrable means that given any \( \epsilon > 0 \), there is a \( \delta > 0 \) such that for every set \( A \) with measure \( \mu(A) < \delta \), the integral \( \int_A |f_n|^p < \epsilon \) holds true.
This concept is crucial in the Vitali Convergence Theorem as it ensures that the integrals of the functions in the sequence do not become excessively large when restricted to subsets of small measure. By controlling the total measure \( \mu(A) \), it becomes possible to manage how much "mass" from \( |f_n|^p \) could "overflow" into \( A \), thereby demonstrating tight control over the sequence's behavior. This control contributes significantly to proving the convergence within the \( L^p \) space.

Measure theory

**Measure theory** is a branch of mathematics that explores concepts relating to size and integration of more abstract sets compared to traditional counting and summation.
It extends the notion of "length" and "area" from geometric figures to more general sets, specifically developed to better understand functions and their behaviors within a more comprehensive scope.
In the context of the Vitali Convergence Theorem, measure theory is used to examine the property "Cauchy in measure."
This setting refers to how the measure of the sets, where functions differ significantly, can be managed to be small. Measure theory thus ensures that even if the functions themselves are not converging pointwise, the differences in measure diminish, showcasing convergence in measure.
Additionally, the description \( \mu(E) < \infty \) and \( \int_{E^c} |f_n|^p < \epsilon \) stems from measure-theoretic concepts. These help in establishing bounds and necessary conditions for managing functions and their integrals beyond simple numeric or finite-dimensional spaces.