Question

Show that if \(S_{1}\) and \(S_{2}\) are Jordan measurable, then so are \(S_{1} \cup S_{2}\) and \(S_{1} \cap S_{2}\).

Short answer

Question: Prove that the union and intersection of two Jordan measurable sets are also Jordan measurable. Short Answer: To prove this, we first find expressions for the boundaries of the union and intersection in terms of the boundaries of the given sets. Then, we use subadditivity of outer measure and the fact that the original sets' boundaries have Jordan measure zero to show that the union and intersection boundaries also have Jordan measure zero. This implies that both the union and intersection of the two Jordan measurable sets are also Jordan measurable.

Step-by-step solution

Finding the boundaries of the sets

Recall that the boundary of a set \(S\) is the set of all points that are limit points of both \(S\) and its complement. We need to find expressions for the boundaries of \(S_{1} \cup S_{2}\) and \(S_{1} \cap S_{2}\) in terms of the boundaries of \(S_{1}\) and \(S_{2}\). By the properties of set operations, we have: \(\partial(S_{1} \cup S_{2}) = (\partial S_{1} \cap \overline{S_{2}}) \cup (\overline{S_{1}} \cap \partial S_{2})\) \(\partial(S_{1} \cap S_{2}) = (\partial S_{1} \cap S_{2}) \cup (S_{1} \cap \partial S_{2})\)

Showing boundary of \(S_{1} \cup S_{2}\) has Jordan measure zero

Since \(S_{1}\) and \(S_{2}\) are Jordan measurable, we know that \(\partial S_{1}\) and \(\partial S_{2}\) have Jordan measure zero. To show that the boundary of \(S_{1} \cup S_{2}\) also has Jordan measure zero, we need to show that the measure of \((\partial S_{1} \cap \overline{S_{2}}) \cup (\overline{S_{1}} \cap \partial S_{2})\) is zero. Since \(\partial S_{1}\) and \(\partial S_{2}\) have Jordan measure zero, they are both of finite outer measure. Therefore, by subadditivity of outer measure: \(0 \leq J^{*}(\partial(S_{1} \cup S_{2})) = J^{*}((\partial S_{1} \cap \overline{S_{2}}) \cup (\overline{S_{1}} \cap \partial S_{2})) \leq J^{*}(\partial S_{1} \cap \overline{S_{2}}) + J^{*}(\overline{S_{1}} \cap \partial S_{2})\) Since the intersection of sets of finite outer measure is also finite, we have: \(J^{*}(\partial S_{1} \cap \overline{S_{2}}) \leq J^{*}(\partial S_{1}) = 0\) \(J^{*}(\overline{S_{1}} \cap \partial S_{2}) \leq J^{*}(\partial S_{2}) = 0\) So, \(J^{*}(\partial(S_{1} \cup S_{2})) = 0\), and hence \(S_{1} \cup S_{2}\) is Jordan measurable.

Showing boundary of \(S_{1} \cap S_{2}\) has Jordan measure zero

Similar to the previous step, we'll show that the measure of \((\partial S_{1} \cap S_{2}) \cup (S_{1} \cap \partial S_{2})\) is zero, which implies that the boundary of \(S_{1} \cap S_{2}\) has Jordan measure zero. Again, using subadditivity of outer measure: \(0 \leq J^{*}(\partial(S_{1} \cap S_{2})) = J^{*}((\partial S_{1} \cap S_{2}) \cup (S_{1} \cap \partial S_{2})) \leq J^{*}(\partial S_{1} \cap S_{2}) + J^{*}(S_{1} \cap \partial S_{2})\) Using the same logic as in Step 2, we have: \(J^{*}(\partial S_{1} \cap S_{2}) \leq J^{*}(\partial S_{1}) = 0\) \(J^{*}(S_{1} \cap \partial S_{2}) \leq J^{*}(\partial S_{2}) = 0\) So, \(J^{*}(\partial(S_{1} \cap S_{2})) = 0\), and hence \(S_{1} \cap S_{2}\) is Jordan measurable. In conclusion, if \(S_{1}\) and \(S_{2}\) are Jordan measurable sets, then their union \(S_{1} \cup S_{2}\) and intersection \(S_{1} \cap S_{2}\) are also Jordan measurable.

Key concepts

Real Analysis

Real analysis is a branch of mathematical analysis dealing with the real numbers and the real-valued functions of a real variable. At its core, it involves the rigorous study of concepts such as convergence, continuity, and the calculation of areas and volumes, which are essential for understanding the properties of functions and spaces in mathematics.

Specifically, real analysis provides the tools to discuss and define measurements, which in this context, refers to the size or 'measure' of a set, particularly how to define and calculate the notion of 'length', 'area', or 'volume'. The exercise pertaining to Jordan measurable sets is squarely within the domain of real analysis, as it directly addresses the concept of measuring the size of particular sets.

Measure Theory

Measure theory is the study of measures, which generalize notions of length, area, and volume in more abstract settings. It provides a systematic way to assign a size to subsets of a given space, typically in the context of the real numbers, with the intent of capturing an intuitive notion of 'how large' a set is.

Measure theory is fundamental in real analysis because it creates a language through which we can discuss the sizes of sets rigorously. This groundwork allows us to explore spaces quite different from simple rectangular areas, such as curvilinear regions, and fractal dimensions, which are not straightforward to measure using traditional Euclidean geometry.

Set Boundaries

The boundary of a set is a fundamental concept in topology, a field closely related to real analysis. In simple terms, the boundary of a set consists of all the points that can be approached both from inside the set and from the outside.

In the context of the Jordan measure, the boundary's importance becomes clear when we assess the size of a set. Sets that are Jordan measurable have boundaries with 'no size', meaning they have a Jordan measure, or 'content', of zero. This idea is integral when dealing with the union and intersection of sets, as addressed in the original exercise, since it pertains to the classification of sets' measurability by their boundaries.

Outer Measure

The 'outer measure' is used to describe the measurement concept applied to all sets, whether or not they are 'measurable' in the traditional sense. It is an extension of the traditional idea of measure designed to handle sets that may not be easily captured by simple geometrical descriptions.

Specifically, in Jordan measure -- which is a precursor to the more general Lebesgue measure -- the outer measure is used to establish whether the boundary of a set is negligible (has measure zero) or not. If a set's outer measure is zero on the boundary, it is Jordan measurable. The steps in the original exercise utilize the concept of outer measure to confirm the measurability of the union and intersection of sets.

Subadditivity

Subadditivity is an essential property in measure theory that relates to the measure of a union of sets. Simply put, it states that the measure of the union of any collection of sets is less than or equal to the sum of their individual measures.

In the exercise provided, subadditivity is crucial for proving that the boundaries of both the union and the intersection of two Jordan measurable sets also have a Jordan measure of zero. The solutions demonstrate the subadditive property by showing that the outer measure of a union of sets (in this case, the boundaries) cannot exceed the sum of the outer measures of the individual sets -- a key factor in determining the Jordan measurability of the resulting sets.