Cauchy In Measure Implies Convergence In Measure

Cauchy in Measure Implies Convergence in Measure: A Detailed Explanation

Meta Description: This article provides a clear and concise proof that a Cauchy sequence in measure converges in measure. We explore the definition of Cauchy sequences and convergence in measure, offering a step-by-step demonstration of the implication. We also discuss the significance of this theorem in measure theory.

In measure theory, understanding the relationship between Cauchy sequences and convergence is crucial. This article will rigorously demonstrate that a sequence of measurable functions that is Cauchy in measure also converges in measure. This is a fundamental result with broad implications in the study of integration and probability.

Understanding the Definitions

Before diving into the proof, let's clearly define the key concepts:

1. Convergence in Measure: A sequence of measurable functions {f_n} converges in measure to a measurable function f on a measurable set E if for every ε > 0,

lim_n→∞ m({x ∈ E : |f_n(x) - f(x)| ≥ ε}) = 0

where m denotes the measure. Essentially, the measure of the set where the difference between f_n and f exceeds ε goes to zero as n approaches infinity.

2. Cauchy in Measure: A sequence of measurable functions {f_n} is Cauchy in measure on a measurable set E if for every ε > 0 and δ > 0, there exists an N such that for all n, m ≥ N,

m({x ∈ E : |f_n(x) - f_m(x)| ≥ ε}) < δ

This means that the measure of the set where the difference between any two functions f_n and f_m (with sufficiently large n and m) exceeds ε can be made arbitrarily small.

Proving the Implication: Cauchy in Measure implies Convergence in Measure

The proof involves constructing a subsequence that converges pointwise almost everywhere, and then showing that the original sequence converges in measure to the same limit.

1. Constructing a Subsequence: Since {f_n} is Cauchy in measure, we can find a subsequence {f_{n_k}} such that for all k ≥ 1,

m({x ∈ E : |f_{n_k+1}(x) - f_{n_k}(x)| ≥ 2^-k}) < 2^-k

This is achieved by recursively choosing the subsequence elements such that the measure of the set where the difference between consecutive terms exceeds 2^-k is less than 2^-k.

2. Pointwise Almost Everywhere Convergence of the Subsequence: Consider the set A_k = {x ∈ E : |f_{n_k+1}(x) - f_{n_k}(x)| ≥ 2^-k}. By the Borel-Cantelli Lemma, since Σ_k=1^∞ m(A_k) < ∞, the measure of the set A = ∪_k=N^∞ A_k goes to zero as N goes to infinity. This means that the subsequence {f_{n_k}} is a Cauchy sequence almost everywhere on E \ A, and thus converges pointwise almost everywhere on E to a function, say f.

3. Convergence in Measure of the Original Sequence: Now, we need to show that the original sequence {f_n} converges in measure to f. Let ε > 0 and δ > 0. Since {f_n} is Cauchy in measure, there exists an N such that for all n, m ≥ N,

m({x ∈ E : |f_n(x) - f_m(x)| ≥ ε/2}) < δ/2

Let m = n_k where n_k is chosen from the subsequence such that n_k ≥ N. Then,

m({x ∈ E : |f_n(x) - f(x)| ≥ ε}) ≤ m({x ∈ E : |f_n(x) - f_{n_k}(x)| ≥ ε/2}) + m({x ∈ E : |f_{n_k}(x) - f(x)| ≥ ε/2}) < δ/2 + δ/2 = δ

for sufficiently large k and n. This demonstrates that the original sequence {f_n} converges in measure to f.

Conclusion

This proof establishes the crucial relationship between Cauchy sequences and convergence in measure. This theorem is a cornerstone in measure theory, providing a foundational tool for analyzing and working with sequences of measurable functions. It underpins many further developments in integration theory and probability. Understanding this theorem is essential for anyone delving deeper into these areas of mathematics.