Question

Let \(f: X \rightarrow \mathbb{R}\) and \(g: X \rightarrow \mathbb{R}\) be mcasurable functicns and \(E \subset X\) a measurable set. Show that $$ h(x)= \begin{cases}f(x) & \text { if } x \in E \\ g(x) & \text { if } x \not E\end{cases} $$ is a measurable function on \(X\).

Short answer

Yes, the piecewise function \(h(x)\) is measurable since the preimage \(h^{-1}(B)\) is measurable for all Borel sets \(B \subseteq \mathbb{R}\).

Step-by-step solution

Understand the definition of a measurable function

Recall that a function \(f: X \rightarrow \mathbb{R}\) is measurable if the preimage \(f^{-1}(B)\) of any Borel set \(B \subseteq \mathbb{R}\) is a measurable set in \(X\). This means that for all \(B \in \mathbb{R}\), if \(f^{-1}(B)\) is measurable in \(X\), then \(f\) is measurable.

Apply the definition to function \(h(x)\)

Given that the functions \(f\) and \(g\) are measurable, for any Borel set \(B \subseteq \mathbb{R}\), both \(f^{-1}(B)\) and \(g^{-1}(B)\) are measurable. We need to show that the preimage \(h^{-1}(B)\) is also measurable.

Divide into two cases

The function \(h(x)\) is a piecewise function that is defined differently for \(x \in E\) and \(x \notin E\). Therefore, the preimage \(h^{-1}(B)\) can be expressed as the union of \(f^{-1}(B) \cap E\) and \(g^{-1}(B) \cap (X \setminus E)\).

Proving \(h^{-1}(B)\) is measurable

Given that \(f^{-1}(B)\) and \(g^{-1}(B)\) are measurable, their intersections with any measurable set are also measurable. Therefore, \(f^{-1}(B) \cap E\) and \(g^{-1}(B) \cap (X \setminus E)\) are measurable since \(E\) and \(X \setminus E\) are measurable. The union of two measurable sets is also a measurable set. Therefore, \(h^{-1}(B)\) is measurable for any Borel set \(B\).

Conclude that \(h(x)\) is measurable

Because \(h^{-1}(B)\) is measurable for all Borel sets \(B\), it follows that the function \(h: X \rightarrow \mathbb{R}\) is a measurable function.

Key concepts

Borel Set

A Borel set is a concept fundamental to the understanding of measure theory and topology. In simple terms, a Borel set is any set of real numbers that can be formed through the operations of countable union, countable intersection, and relative complement, starting with the open intervals in \( \mathbb{R} \). This means that Borel sets include open sets, closed sets, and any countably infinite combination of these basic sets.
Borel sets are crucial because they serve as the building blocks for more complex sets and are particularly important in defining measurable sets. They ensure that when we talk about the "measurability" of functions and sets, we are referencing a structure that is well-understood and operates within the framework of standard Euclidean spaces. In the context of the exercise, we deal with Borel sets to determine the measurability of the given piecewise function.

Piecewise Function

A piecewise function is a type of function defined by different expressions depending on the input's value. It is often symbolized as having distinct clauses that apply to different parts of its domain, much like breaking a task into smaller, manageable parts.
In the exercise problem, the function \( h(x) \) is defined piecewise. It takes the form \( f(x) \) when \( x \) belongs to the set \( E \), and \( g(x) \) when \( x \) does not belong in \( E \). Using piecewise functions allows for flexibility in defining functions, letting them change behavior based on certain logical conditions. They are widely used in mathematics to describe behaviors that can change abruptly, such as absolute value functions or complex-shaped graphs in physics or engineering.

Preimage

The preimage is a fundamental concept in understanding functions and transformations in mathematics. It refers to the set of all input values which map to a given output or set of outputs through the function.
Mathematically, the preimage of a set \( B \) under a function \( f: X \rightarrow Y \) is denoted as \( f^{-1}(B) \), which includes all elements \( x \in X \) such that \( f(x) \in B \).
In solving the exercise, the concept of preimage helps us understand how a measurable set can be built using measurable functions. For the piecewise function \( h(x) \), we consider the preimages \( f^{-1}(B) \cap E \) and \( g^{-1}(B) \cap (X \setminus E) \). These form the basis for proving the measurability of the function \( h(x) \).

Measurable Set

Measurable sets are essential in the study of measure theory, playing a key role in defining what it means for a function to be measurable. A set is measurable if it fits within the structure defined by a sigma-algebra, commonly based on Borel sets in Euclidean spaces.
Measurable sets allow mathematicians to extend the notion of "length" or "volume" to more complex sets, making integration and probability theory workable.

• For instance, the preimage of a Borel set under a measurable function is always measurable.
• This ensures functions like \( f(x) \) and \( g(x) \) from the exercise, given as measurable, can be broken into measurable sections when intersected with \( E \) or \( X \setminus E \).
• The union of two measurable sets is also measurable, which is why the structure of \( h^{-1}(B) \) becomes coherent and measurable.

Understanding the properties of measurable sets allows us to delve deeper into analysis and address complex problems in a structured way, just like in the provided exercise where the piecewise function is shown to be measurable.