Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma

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Theorem

Let $\closedint a b$ be a closed real interval.

Let $c$ be a real number.

Let $a < c < b$.


Let $f$ be a real function defined on $\closedint a b$.

Let $\map L S$ be the lower Darboux sum of $f$ on $\closedint a b$ where $S$ is a subdivision of $\closedint a b$.


Let $P$ and $Q$ be finite subdivisions of $\closedint a b$.

Let:

$Q = P \cup \set c$.


Then:

$\map L Q \ge \map L P$


Proof 1

This is an instance of Lower Sum of Refinement.

$\blacksquare$


Proof 2

Let $P = \set {x_0, x_1, \ldots, x_n}$.


Suppose that:

$c \in P$

Then:

$Q = P$

We have:

$\map L P \ge \map L P$
$\leadsto \map L Q \ge \map L P$ as $Q = P$

This finishes the proof for this case.


The only other possibility for $c$ is:

$x_{j-1} < c < x_j$

where $1 \le j \le n$.

Let $m_i$ be the infimum of $f$ on the interval $\closedint {x_{i - 1}} {x_i}$ for all $i \in \set {1, 2, \ldots, n}$.

We have:

\(\ds \map L P\) \(=\) \(\ds \sum_{i \mathop = 1}^n m_i \paren {x_i - x_{i - 1} }\) Definition of Lower Darboux Sum
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j − c + c - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j − c} + m_j \paren {c - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {c - x_{j - 1} } + m_j \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)

Let $r$ be the infimum of $f$ on the interval $\closedint {x_{j - 1}} c$.

Let $s$ be the infimum of $f$ on the interval $\closedint c {x_j}$.


The interval $\closedint {x_{j - 1}} c$ is a subset of $\closedint {x_{j - 1}} {x_j}$.

Therefore, a lower bound of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound of $f$ on the interval $\closedint {x_{j - 1}} c$ as well.


The infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound for $f$ on this interval.

Accordingly, the infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound of $f$ on the interval $\closedint {x_{j - 1}} c$ as well.

Therefore, the infimum of $f$ on the interval $\closedint {x_{j - 1}} c$ is greater than or equal to the infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$.

In other words:

$r \ge m_j$


A similar analysis of $\closedint c {x_j}$ gives:

$s \ge m_j$


We find:

\(\ds \map L P\) \(=\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {c - x_{j - 1} } + m_j \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)
\(\ds \) \(\le\) \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + r \paren {c - x_{j - 1} } + s \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\)
\(\ds \) \(=\) \(\ds \map L Q\) Definition of Lower Darboux Sum

This finishes the proof.

$\blacksquare$