Riemann Integral, Riemann Sums, & Improper Riemann Integration - Introduction to Analysis

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Question

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Answer

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where represents the upper integral and the following are defined:

1. The upper integral of on is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ where is a partition of .

2. The lower integral of on is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where is a partition of .

3. If 1 and 2 are the same then the integral is said to be

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if , $b\ \epsilon\ \mathbb{R}$ , , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , $b\ \epsilon\ \mathbb{R}$ , , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

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Question

What term has the following definition.

, $b\ \epsilon\ \mathbb{R}$ and . Over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

Answer

By definition

If , $b\ \epsilon\ \mathbb{R}$ and .

A partition over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

.

Therefore, the term that describes this statement is partition.

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Question

What term has the following definition.

The __________ of a partition $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Answer

By definition

If , $b\ \epsilon\ \mathbb{R}$ and .

A partition over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

.

Furthermore,

The norm of the partition

$P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Therefore, the term that describes this statement is norm.

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