Question

Suppose \(f, g: X \rightarrow \mathbb{R}\) are measurable. a. \(f g\) is measurable (where \(0 \cdot(\pm \infty)=0\) ). b. Fix \(a \in \overline{\mathbb{R}}\) and define \(h(x)=a\) if \(f(x)=-g(x)=\pm \infty\) and \(h(x)=\) \(f(x)+g(x)\) otherwise. Then \(h\) is measurable.

Short answer

Both \( f g \) and \( h \) are measurable.

Step-by-step solution

Understand measurability

A function is measurable if the preimage of every Borel set is a measurable set in \( X \). This is a fundamental property when dealing with functions that map from a measurable space.

Show that product of measurable functions is measurable

Given measurable functions \( f \) and \( g \), the product \( fg \) is measurable. This is because the product of two measurable functions is known to be measurable in the realm of real-valued functions. Additionally, \( 0 \cdot (\pm \infty) = 0 \) ensures the product is well-defined wherever \( f \) or \( g \) may be infinite.

Define the function h based on conditions

We have the function \( h(x) = a \) if \( f(x) = -g(x) = \pm \infty \) and \( h(x) = f(x) + g(x) \) otherwise. The function is separately defined based on these conditions.

Analyze measurability of h based on the conditions

To show \( h \) is measurable, check each condition separately. Since \( f \) and \( g \) are measurable, \( f + g \) is measurable as the sum of measurable functions is measurable. For the condition \( f(x) = -g(x) = \pm \infty \), the set where this occurs must also be measurable because \( f \) and \( g \) are functions from a measurable space and their respective infinities would represent measurable locus.

Conclude measurability of h

Given that each part of the definition of \( h \) is measurable based on the conditions, \( h \) itself is measurable. This is because a function defined piecewise by measurable conditions over measurable subsets of \( X \) is measurable.

Key concepts

Borel Sets

Borel sets play a central role in real analysis as they form the smallest collection of sets for which we can define a meaningful notion of measure. A Borel set is constructed from open intervals in the real numbers through the operations of countable union, countable intersection, and relative complement. These sets effectively pave the way for defining concepts such as measurable spaces and functions.

In real analysis, we often deal with the \"Borel \sigma-algebra\", which is an algebra of sets generated by Borel sets. This is crucial because in order for a function to be measurable, its domain must be a measurable space, often equipped with the Borel \sigma-algebra.

• Borel sets can include intervals like \((a, b)\), \([a, b)\), or \(\{a\}\).
• Any set derived from open sets through the mentioned operations falls under Borel sets.
• The notion of Borel sets extends beyond just real numbers to more general spaces, facilitating complexes studies like integration.

This concept serves as a foundational block upon which the structure of real analysis is built, including the study of measurable spaces and functions.

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and real-valued sequences and functions. Comprehending real analysis is essential for understanding advanced mathematical constructs and theories. Its focus is on rigorously handling limits, continuity, differentiation, and integration, using the foundational framework of sets and measure theory.

In real analysis, measurable functions, such as those discussed in the exercise, highlight an aspect of analysis that deals with functions having certain predictability over "measurable" domains.

• Real analysis provides tools to handle infinitely small and large quantities through limits and measure.
• A measurable function is one which is compatible with the structure of the measurable space it is defined on.
• Key theorems, like the Dominated Convergence Theorem and Fubini’s Theorem, rely on the understanding of measurable sets and functions.

The exercise involves key real analysis principles—specifically, how functions behave within measurable constructs and what types of operations retain measurability.

Measurable Space

A measurable space is a foundational concept in measure theory, which is a vital aspect of real analysis. A measurable space is a set equipped with a \(\sigma\)-algebra, usually composed of Borel sets, allowing the study of measurable functions and integration.

The concept of a measurable space is crucial when determining the measurability of a function. When the measurable functions from the exercise map from one measurable space to another, they preserve the structure necessary for measuring, integrating, and analyzing the functions.

• A measurable space consists of a set and a collection of its subsets, the \(\sigma\)-algebra.
• It allows the formal definition and operation over sets, serving as the domain or codomain for measurable functions.
• Through the\(\sigma\)-algebra, one can talk about events, outcomes, and probability in more formal terms in spaces.

This concept ties back to the essence of ensuring that operations on functions lead to measurable outputs, which is critical in applications across probability and real analysis. Understanding measurable spaces helps understand how functions can retain certain properties post-operations, like those seen in sum and product scenarios within the exercise.