Question

If \(X=A \cup B\) where \(A, B \in \mathcal{M}\), a function \(f\) on \(X\) is measurable iff \(f\) is measurable on \(A\) and on \(B\).

Short answer

\(f\) is measurable on \(X = A \cup B\) if and only if \(f\) is measurable on \(A\) and on \(B\).

Step-by-step solution

Understanding the Problem

We are given a set \(X = A \cup B\) where \(A\) and \(B\) are measurable sets. We have a function \(f\) defined on \(X\) and need to show that \(f\) is measurable on \(X\) if and only if it is measurable on both \(A\) and \(B\).

Definition of Measurable Function

A function \(f\) is measurable on a set \(E\) if for every real number \(c\), the set \(\{ x \in E \mid f(x) > c \}\) is measurable. We will use this definition for sets \(A\), \(B\), and \(X = A \cup B\).

Proving Necessity

Assume \(f\) is measurable on \(X\). For \(A \subseteq X\) and \(B \subseteq X\), measurability of \(f\) on \(X\) implies measurability on any subset, so \(\{ x \in A \mid f(x) > c \}\) and \(\{ x \in B \mid f(x) > c \}\) are measurable for every real \(c\). Thus, \(f\) is measurable on both \(A\) and \(B\).

Proving Sufficiency

Assume \(f\) is measurable on \(A\) and \(B\). For every real \(c\), the sets \(\{ x \in A \mid f(x) > c \}\) and \(\{ x \in B \mid f(x) > c \}\) are measurable. Since \(\{ x \in X \mid f(x) > c \} = \{ x \in A \mid f(x) > c \} \cup \{ x \in B \mid f(x) > c \}\), and the union of measurable sets is measurable, \(f\) is measurable on \(X\).

Conclusion

A function \(f\) on \(X = A \cup B\) is measurable if and only if it is measurable on both \(A\) and \(B\).

Key concepts

Measurable Sets

Measurable sets are an essential concept in real analysis. They are the building blocks for defining measurable functions. A measurable set is one that can be assigned a meaningful "size" or "volume," usually with respect to a particular measure, such as Lebesgue measure in real analysis. The concept is critical in integration theory and probability. Essentially, a set is measurable if it behaves nicely with respect to the measure, allowing for the accurate evaluation and integration over the set.

To be more specific, a set is termed measurable if it belongs to a σ-algebra associated with a measure. This often means that complex sets can be built through operations like countable unions, intersections, and complements. Hence, measurable sets allow integration and summation to extend beyond simple cases. Understanding measurable sets is crucial for analyzing the properties of functions, particularly when dealing with measurable functions.

Union of Sets

The union of sets is a fundamental operation in set theory, crucial for understanding measurable functions in real analysis. The union of two sets, denoted by \(X = A \cup B\), consists of elements that are in either set \(A\), set \(B\), or both. This operation helps combine multiple sets into a single entity, making it easier to analyze or measure the combined set.

When dealing with measurable sets, the union process preserves measurability. If \(A\) and \(B\) are both measurable, their union \(A \cup B\) is also measurable. This property is vital when working with measurable functions defined over these sets. The reasoning behind this is based on the closure properties of the σ-algebra that define measurable sets, ensuring any countable union of measurable sets remains measurable. This concept underlies the ability to analyze more complex functions that might be defined over united districts.

Real Analysis

Real analysis provides the foundation for much of the rigorous treatment of calculus and introduces a formal framework to understand concepts like continuity, integration, and differentiation. Central to real analysis is the study of real-valued measurable functions and their properties. In this context, a measurable function maps measurable sets to real numbers in a way that respects the underlying measure structure.

One of the key aspects of real analysis is understanding how these functions behave under various set operations, such as unions and intersections. The ability to determine whether a function is measurable based on its restrictions to measurable sets, like \(A\) and \(B\), is an example of this framework. Real analysis extends calculus by providing the techniques needed to manage diverse situations, including handling potential discontinuities and unbounded regions with rigorously defined limits and integrals. This deeper insight is essential for advanced applications in mathematics, physics, and engineering, particularly when solving complex problems involving continuous change.