Question

Prove that if \(\int f\) exists on each \(I_{n}=\left[a_{n}, b_{n}\right],\) where $$ a_{n+1} \leq a_{n} \leq b_{n} \leq b_{n+1}, \quad n=1,2, \ldots $$ then \(\int f\) exists on $$ I=\bigcup_{n=1}^{\infty}\left[a_{n}, b_{n}\right] $$ itself an interval with endpoints \(a=\inf a_{n}\) and \(b=\sup b_{n}, a, b \in E^{*}\). [Hint: Fix some \(c \in I_{1}\). Define $$ H_{n}(t)=\int_{c}^{t} f \text { on } I_{n}, n=1,2, \ldots $$ Prove that $$ (\forall n \leq m) \quad H_{n}=H_{m} \text { on } I_{n}\left(\text { since }\left\\{I_{n}\right\\} \uparrow\right) . $$ Thus \(H_{n}(t)\) is the same for all \(n\) such that \(t \in I_{n},\) so we may simply write \(H\) for \(H_{n}\) on \(I=\bigcup_{n=1}^{\infty} I_{n} .\) Show that \(H=\int f\) on all of \(I ;\) verify that \(I\) is, indeed, an interval.]

Short answer

The existence of \(\int f\) on each \(I_n\) implies its existence on \(I\). \(I\) is an interval given its sequential endpoint definitions.

Step-by-step solution

Understand the Interval Structure

Given the sequence of intervals \(I_n = [a_n, b_n]\), they are nested such that \(a_{n+1} \leq a_n \leq b_n \leq b_{n+1}\). The larger interval \(I\) is defined as the union of all these intervals: \(I = \bigcup_{n=1}^{\infty}[a_n, b_n]\), with endpoints \(a = \inf a_n\) and \(b = \sup b_n\). This means as we progress through the sequence, the intervals expand or remain the same, covering all points from \(a\) to \(b\).

Defining the Intermediary Function

For each interval \(I_n\), define \(H_n(t) = \int_{c}^{t} f\) where \(c \in I_1\). This function represents the integral of \(f\) from a fixed point \(c\) to a variable endpoint \(t\) within each interval \(I_n\).

Demonstrate Consistency Across Intervals

Show that for any two intervals \(I_n\) and \(I_m\) with \(n \leq m\), the function \(H_n(t)\) equals \(H_m(t)\) on \(I_n\). Since the sequence of intervals is increasing \(\{I_n\} \uparrow\), the integral from \(c\) to \(t\) should be the same for both intervals as long as \(t \in I_n\).

Define \(H\) on the Union of Intervals

Since \(H_n(t)\) is defined consistently across all intervals, we can define \(H(t) = H_n(t)\) for all \(t\) in \(I = \bigcup_{n=1}^{\infty} I_n\). This defines \(H\) on the entire interval \(I\).

Verify \(H = \int f\) on \(I\)

Since \(H(t)\) is defined as an integral from \(c\) to \(t\) for all \(t\) in \(I\), \(H\) is indeed the integral of \(f\) over the whole interval \(I\). This shows that the integral \(\int f\) exists over \(I\), as \(H\) correctly represents the area under \(f\) from \(c\) to \(t\).

Confirm \(I\) is an Interval

The fact that \(a = \inf a_n\) and \(b = \sup b_n\) ensures \(I = \bigcup_{n=1}^{\infty} [a_n, b_n]\) contains all real number points between \(a\) and \(b\). This property confirms that \(I\) is indeed an interval in the real number system.

Key concepts

Interval Structure

The structure of intervals in the context of integrals is fundamental to understanding how they contribute to the existence of an integral over a larger set. In this exercise, the sequence of intervals, denoted by \(I_n = [a_n, b_n]\), forms a nested arrangement. This means that each subsequent interval is either expanded or equal to the previous one with respect to their endpoints: \(a_{n+1} \leq a_n \leq b_n \leq b_{n+1}\). As we move through this sequence, the intervals cover all points between \(a\) and \(b\). The critical aspect here is that the intervals are increasing, allowing for the definition of a larger interval \(I\) as the union of all smaller intervals. This ensures the inclusion of every point between \(a = \inf a_n\) and \(b = \sup b_n\), firmly establishing \(I\) as an interval in the real number system.

Intermediary Function

In the process of proving the existence of an integral over a union of intervals, we define an intermediary function \(H_n(t)\). For each interval \(I_n\), this function is expressed as \(H_n(t) = \int_{c}^{t} f\), where \(c\) is a fixed point within the initial interval \(I_1\).
The function \(H_n(t)\) effectively represents the integral of the function \(f\) from a constant starting point \(c\) to any variable endpoint \(t\) within interval \(I_n\). This step is crucial as it helps in understanding how the integral behaves across different segments of our larger interval. By creating \(H_n(t)\), we lay the groundwork for checking its consistency across various intervals, leading to the seamless integration over the entire union \(I\).

Consistency Across Intervals

Consistency across intervals signifies that the intermediary function is identical over overlapping regions of consecutive intervals. To establish this, we show that for any \(n\) and \(m\) such that \(n \leq m\), the functions \(H_n(t)\) and \(H_m(t)\) are equivalent wherever they are both defined, particularly on \(I_n\).
This is possible because the sequence \(\{I_n\}\) ascends, meaning each subsequent interval either encompasses or is equal to the previous, allowing the function \(H_n(t)\) to remain invariant if \(t\) is within \(I_n\). This is a pivotal point in proving that the function can be extended throughout the entire interval \(I\) by establishing that it doesn't fluctuate across adjoining intervals.

Union of Intervals

The union of intervals is a process that combines an infinite sequence of overlapping intervals into a larger, continuous interval \(I\). In mathematical terms, \(I = \bigcup_{n=1}^{\infty} [a_n, b_n]\) means that we gather all elements from each interval to form a unified set.
With the definition of \(H_n(t)\) showing consistency across intervals, the function can be considered as \(H(t)\), valid for all points within \(I\). This substantiates that \(H(t)\) is synonymous with the integral of the function across \(I\), as it remains unchanged over the elements drawn from each \(I_n\). As such, the notion of the union ensures that every real number point between \(a\) and \(b\) is captured, verifying \(I\) as a bona fide interval.

Real Analysis

Real analysis provides the foundational tools necessary to tackle problems involving limits, integration, and continuous functions over the real numbers. This particular exercise is grounded in real analysis by exploring how the properties of intervals and functions define an integral across an enlarged set of values.
Here's how real analysis concepts come into play:

• Limit properties: The endpoints, \(a\) and \(b\), set as infimum and supremum, demonstrate key concepts of bounds and completeness.
• Continuous functions: The defined intermediary function \(H(t)\) utilizes continuity across intervals to extend its validity over a union.
• Integrals: The consistent integration of \(f\) over each \(I_n\) reflects the power of defining an integral via limits.

By examining these aspects, students can grasp how real analysis principles support the formulation and resolution of complex integration tasks, ultimately confirming the existence and behavior of integrals over extended intervals.