Question

Let \((X, \mathcal{M}, \mu)\) be a measure space with \(\mu(X)<\infty\), and let \((X, \overline{\mathcal{M}}, \bar{\mu})\) be its completion. Suppose \(f: X \rightarrow \mathbb{R}\) is bounded. Then \(f\) is \(\overline{\mathbb{M}}\)-measurable (and hence in \(\left.L^{1}(\bar{\mu})\right)\) iff there exist sequences \(\left\\{\phi_{n}\right\\}\) and \(\left\\{\psi_{n}\right\\}\) of \(\mathcal{M}\)-measurable simple functions such that \(\phi_{n} \leq f \leq \psi_{n}\) and \(\int\left(\psi_{n}-\phi_{n}\right) d \mu

Short answer

The function \(f\) is \(\overline{\mathcal{M}}\)-measurable iff it can be approximated by \(\mathcal{M}\)-measurable simple functions sequences.

Step-by-step solution

Understand the Measure Space

We start with a measure space \((X, \mathcal{M}, \mu)\), where \(\mu(X)<\infty\), meaning the measure of the whole space \(X\) is finite. The completion of this measure space is denoted by \((X, \overline{\mathcal{M}}, \bar{\mu})\), which means it includes all subsets of measure zero sets.

Define Bounded Function

The function \(f: X \rightarrow \mathbb{R}\) is assumed to be bounded, meaning there exists some constant \(M\) such that \(|f(x)| \le M\) for all \(x \in X\). This boundedness is important for integration and approximation.

Define Simple Functions

Simple functions are functions that take a finite number of values and are measurable. The sequences \(\{\phi_n\}\) and \(\{\psi_n\}\) will be these simple functions that approximate \(f\) from below and above, respectively.

Measurable Functions Condition

The condition \(\phi_n \leq f \leq \psi_n\) ensures that \(f\) is squeezed between these simple functions. This is necessary for exploring the measurability of \(f\).

Integration Difference Condition

The condition \(\int(\psi_n - \phi_n) \, d\mu < n^{-1}\) implies that the simple functions \(\phi_n\) and \(\psi_n\) are converging sufficiently closely to \(f\). This ensures that these functions approximate \(f\) well as \(n\) increases.

Prove Measurability Implication

Given the conditions, \(f\) is shown to be \(\bar{\mathcal{M}}\)-measurable. This is because we can approximate \(f\) sufficiently closely using \(\mathcal{M}\)-measurable simple functions, which are dense in \(L^1(\mu)\).

Evaluate Integrals

Since \(f\) is approximated closely, the integral \(\lim \int \phi_n \, d\mu = \lim \int \psi_n \, d\mu = \int f \, d\mu\) holds. The integrals of the sequences converge to the integral of \(f\), corroborating our previous steps.

Key concepts

Bounded Functions

In measure theory, a function is considered bounded if there exists a real number, say \( M \), such that \( |f(x)| \leq M \) for all elements \( x \) in the set \( X \). This quality of boundedness plays a crucial role in the analysis and integration of these functions, especially within measure spaces.Bounded functions are critical when discussing the convergence and approximation of functions. The boundedness ensures that the function doesn't "explode" to infinity, making it much easier to handle mathematically.When a function is bounded, it is possible to estimate it with simple functions from above and below. This ability to "trap" the bounded function allows us to work effectively within the space of integrable functions, as boundedness often links directly to measurability.

Simple Functions

Simple functions are an essential concept in measure theory. They are those that only take on a finite number of distinct values and are measurable with respect to a given \( \sigma \)-algebra. Essentially, simple functions can be considered as building blocks or approximations of more complex functions.

Characteristics of Simple Functions

• They are measurable, meaning they respect the structure of the measure space.
• They map every point in their domain to a finite set of real numbers.
• They can be used to approximate other more complex functions by being squeezed, or sandwiched, between them.

The role of simple functions is pivotal as they serve as approximations within integrations. In the specified problem, the sequences \( \{\phi_n\} \) and \( \{\psi_n\} \) are simple functions, acting to approximate the bounded function \( f \) from below and above, respectively. As \( n \) increases, the simple functions become closer to \( f \), hinting at the function’s measurability by capturing its finer details.

Measurable Functions

A function is termed measurable if it aligns with a particular \( \sigma \)-algebra associated with a measure space. For any set \( A \) in the real numbers, the pre-image of \( A \) under a measurable function is a measurable set in the space under consideration.

Importance of Measurability

• Measurability ensures that we can integrate the function over a given measure space.
• It establishes the function's compatibility with the structure of the measure space, allowing rigorous analysis of its properties.
• Helps manage complex functions through simpler and more understandable entities.

Being \( \overline{\mathcal{M}} \)-measurable implies that the function can be approximated arbitrarily closely by bouts of simple functions. In the context of the problem, measurability is verified through the approximation of \( f \) with sequences of simple \( \mathcal{M} \)-measurable functions. This way, measurable functions align with the dominant frameworks of measure theory, laying the foundations for further applied analysis.