Question

\(\operatorname{lf}(X, M)\) and \((Y, N)\) are measurable spaces, show that the projection maps \(\mathrm{pr}_{1}: X \times\) \(Y \rightarrow X\) and \(\mathrm{pr}_{2}: X \times Y \rightarrow Y\) defined by \(\mathrm{pr}_{1}(x, y)=x\) and \(\mathrm{pr}_{2}(x, y)=y\) are measurable functions,

Short answer

Both projection maps \(\mathrm{pr}_{1}\) and \(\mathrm{pr}_{2}\) are measurable functions, as their pre-images map any measurable set to a measurable set in \(X \times Y\).

Step-by-step solution

Verify Definition of Measurable Function

A function is measurable if the pre-image of any measurable set under the function is also a measurable set. Let's prove \(\mathrm{pr}_{1}: X \times Y \rightarrow X\) and \(\mathrm{pr}_{2}: X \times Y \rightarrow Y\) are measurable by this definition.

Analyze Projection Map \(\mathrm{pr}_{1}\)

Take a set \(A \subseteq X\) that is \(\sigma\)-measurable. The pre-image of \(A\) under the function \(\mathrm{pr}_{1}\) is given by the set \(\mathrm{pr}_{1}^{-1}(A) = A \times Y\). As \(A\) is a measurable set and \(Y\) is a complete space, their Cartesian product is a measurable set in the product \(\sigma\)-algebra on \(X \times Y\). Therefore, for any measurable set \(A \subseteq X\), \(\mathrm{pr}_{1}^{-1}(A)\) is also measurable, so \(\mathrm{pr}_{1}\) is a measurable function.

Analyze Projection Map \(\mathrm{pr}_{2}\)

Similarly, for a set \(B \subseteq Y\) that is \(\sigma\)-measurable, the pre-image of \(B\) under the function \(\mathrm{pr}_{2}\) is given by the set \(\mathrm{pr}_{2}^{-1}(B) = X \times B\). As \(B\) is a measurable set and \(X\) is a complete space, their Cartesian product is a measurable set in the product \(\sigma\)-algebra on \(X \times Y\). Therefore, for any measurable set \(B \subseteq Y\), \(\mathrm{pr}_{2}^{-1}(B)\) is also measurable, so \(\mathrm{pr}_{2}\) is a measurable function.

Key concepts

Measurable Spaces

In mathematics, the concept of measurable spaces forms the foundation for the study of measure theory. A measurable space is made up of a set, which is often called the universe of discourse, and a collection of subsets, which form a sigma-algebra. The sigma-algebra, denoted as \(\sigma\)-algebra, is a collection of subsets closed under complementation and countable unions.

This means that if you have a \(\sigma\)-algebra \(M\) over a set \(X\), you can form any subset from the available measurable subsets through operations like taking complements and countable unions.
- **Set**: Provides the domain.- **Sigma-Algebra**: Defines measurable events.
Measurable spaces are crucial because they allow us to assign measures—think of measures as generalized notions of size or probability—to sets in a rigorous way. The structure ensures that important properties like countable additivity are maintained, making various sets measurable and suitable for analysis.

Projection Maps

Projection maps are basic functions that help us examine more complex structures by focusing on one component at a time. Specifically, they project a product space onto one of its factor spaces.

Let's consider a Cartesian product \(X \times Y\). The projection map \(\mathrm{pr}_1 : X \times Y \rightarrow X\) takes a pair \((x, y)\) and projects it to \(x\), discarding the \(y\) component. Likewise, \(\mathrm{pr}_2 : X \times Y \rightarrow Y\) projects \((x, y)\) to \(y\).
- **First Projection** \(\mathrm{pr}_1\): Focuses on the first element \(x\).- **Second Projection** \(\mathrm{pr}_2\): Focuses on the second element \(y\).
An important property of these projection maps is measurability. A projection is measurable if the pre-image of any measurable set is also measurable. This is crucial for managing and analyzing data in higher-dimensional spaces.

Product σ-algebra

The product \(\sigma\)-algebra is a construction that allows us to define a measurable structure on the Cartesian product of measurable spaces. When you have two measurable spaces, say \((X, M)\) and \((Y, N)\), their product \(\sigma\)-algebra \(M \otimes N\) helps in studying functions and products that span both spaces.

This product \(\sigma\)-algebra is generated by rectangles \(A \times B\) where \(A\) belongs to \(M\) and \(B\) belongs to \(N\). This ensures that not only the individual spaces are measurable, but the combination of these spaces also retains measurability.

• **Closure Properties**: Similar to individual \(\sigma\)-algebras.
• **Generation by Rectangles**: The smallest \(\sigma\)-algebra that includes all sets of the form \(A \times B\).

Understanding the product \(\sigma\)-algebra is essential for dealing with multivariate data, where one deals with observations from multiple dimensions at the same time.