Question

Suppose that \(H\) is a subset of a compact Jordan measurable set \(S\) such that the intersection of \(H\) with any compact subset of \(S^{0}\) has zero content. Show that \(V(H)=0\).

Short answer

#Question# Show that the volume of a subset H of a compact Jordan measurable set S is zero if and only if the intersection of H with any compact subset of the interior of S (S^0) has zero content.

Step-by-step solution

Define the volume function and content of a set

The volume function \(V(H)\) is defined as the integral of the characteristic function of \(H\) over its region, i.e., \(V(H) = \iiint_R \chi_H(x,y,z) \,dx\,dy\,dz\), where \(R\) is the region that contains the set \(H\). The content of a set is measured by its volume, so if a set has zero content, its volume is zero.

Use the assumption

By the assumption, the intersection of \(H\) with any compact subset of \(S^0\) has zero content. Let \(C\) be any compact subset of \(S^0\). Then, \(V(H\cap C) = 0\).

Consider the union of \(H\) and compact subset \(C\)

Since \(H \subset S\) and \(S\) is Jordan measurable, we can consider the Jordan measurable set \(S' = H \cup C\). It follows that \(H \cap C\) has zero content. Moreover, \(H\) is a subset of \(S'\), and since the interior of a set is disjoint from its boundary, we have \(H\cap \partial S' = \emptyset\).

Find an upper bound for the volume of \(H\)

To find an upper bound for the volume of \(H\), observe that for any \(\epsilon > 0\), there exists a compact subset \(C_\epsilon\) of \(S^0\) such that \(V(H\cap C_\epsilon) < \epsilon\). Then we can write the set \(H\) as the disjoint union \(H = (H\cap C_\epsilon) \cup (H\cap C_\epsilon^c)\). Since volume is additive for disjoint sets, we have \(V(H) \leq V(H\cap C_\epsilon) + V(H\cap C_\epsilon^c) < \epsilon + V(H\cap C_\epsilon^c)\).

Show that \(V(H)=0\)

Since \(H\cap C_\epsilon\) has zero content and \(\epsilon > 0\) is arbitrary, \(V(H\cap C_\epsilon) = 0\). Therefore, \(V(H) \leq \epsilon + V(H\cap C_\epsilon^c) = V(H\cap C_\epsilon^c)\). As this inequality holds true for any compact subset \(C_\epsilon\) of \(S^0\), it must be true that \(V(H) \leq 0\). But volume cannot be negative, so \(V(H) = 0\). Thus, we have shown that the volume of the set \(H\) is zero, implying that it has zero content.

Key concepts

Compact Subsets

In mathematics, a subset is called compact if it holds particular useful properties, especially in the context of analysis. For a set to be compact, it must be closed and bounded. This essentially means that every point in the set can be enclosed within some region of finite measure, and the points are also contained in the boundary of that set.

When dealing with compact subsets, especially in terms of Jordan measurable sets, it's significant because these subsets maintain order and limit, making it easier to apply certain mathematical principles like limits and continuity. This property is robust enough that when applied to Jordan measurable sets, which are those sets that can be well-approximated by simpler measurable sets, it provides valuable insights for solving problems related to volumes and measures.

Compact subsets play a critical role in proving certain properties of the set in focus, specifically the fact that the intersection of any compact subset with another related set might have inherently controlled characteristics like zero measure (content). This can simplify evaluations, especially when proving concepts like having zero volume, as it happens in our exercise.

Zero Content

The concept of zero content is crucial when discussing the properties of sets in analysis, especially in Jordan measurable contexts. A set is said to have zero content if its measure, or volume in an intuitive sense, is zero. In practical terms, it implies that the set occupies no space, even though it might contain infinitely many points.

In the provided exercise, the intersection of subset \(H\) with any compact subset of \(S^0\) having zero content is a crucial assumption that is used to prove the zero content of \(H\). By definition, since any intersection with a compact subset of the interior of \(S\) (denoted by \(S^0\)) has zero content, it allows us to assert that the whole of \(H\) also has zero content. The zero content acts as a powerful tool in showing certain mathematical properties about spaces and sets, compressing into manageable information that a set assumes no measurable space, which aids in proving its volume is zero.

This leads to the conclusion that for any sequence of compact subsets, the contained smaller intersections also reflect the overall property of being zero, ensuring the whole set in question inherits this property.

Volume Function

The volume function is a fundamental concept in integration and measure theory, which assigns a real number representing the measure of a set. Specifically for three-dimensional spaces, it corresponds to the integral of the set's characteristic function over its entire domain. The characteristic function, \(\chi_H(x,y,z)\), is 1 for elements inside the set \(H\) and 0 outside it.

In the problem scenario, the volume function \(V(H)\) calculates the measure of the set \(H\) and is an indication of the 'size' of \(H\) in terms of space it inhabits. The problem requires proving that this function is zero for the given conditions, showing that no actual volume is occupied.

• The integral representation: \(V(H) = \iiint_R \chi_H(x,y,z) \,dx\,dy\,dz\), sums up all infinitesimal elements covered by \(H\).
• Since every intersection \(H\cap C\) holds zero volume, the function sums correspondingly to confirm zero volume across all \(H\).

Understanding this function aids in grasping the abstract yet powerful concept that even a set \(H\) enclosed within other measurable spaces might share such properties. Consequently, it's pivotal in proving important results in analysis and measure theory.