# 数学代写|数学分析代写Mathematical Analysis代考|Properties of Riemann Integrals

## 数学代写|数学分析代写Mathematical Analysis代考|Properties of Riemann Integrals

This section aims at establishing reduction formulas, using which the integration of a bounded function over a measurable bounded set in $\mathbb{R}^{n+1}$ can be reduced to an (iterated) integral of functions of one variable.

We shall need some definitions, that generalise the analogous definitions in two and three dimensions for normal domains seen in the first part of the chapter. Let $\alpha=\alpha(x)$ and $\beta=\beta(x)$ be continuous functions on a closed, bounded and measurable set $X$ of $\mathbb{R}^n$, and such that
$$\alpha(x) \leq \beta(x), \quad \forall x \in X$$
The subset of $\mathbb{R}^{n+1}$
$$D={(x, y) \in X \times \mathbb{R}: \alpha(x) \leq y \leq \beta(x)}$$

will be called normal set (with respect to the first $n$ coordinates) over $X$. Analogous notions exist for any collection of $n$ coordinates.

Measurability of Normal Sets Let $X \subset \mathbb{R}^n$ be closed, bounded and measurable, and $D$ the normal set over $X$ defined in (8.98). Then $D$ is measurable and its measure is
$$m_{n+1}(D)=\int_X{\beta(x)-\alpha(x)} d x$$
Proof First we consider the case $\alpha(x) \equiv 0$. Let $P=\left{X_1, X_2, \ldots, X_h\right}$ be a partition of $X$ into measurable sets. Correspondingly,
$$A_i=X_i \times\left[0, \inf \beta\left(X_i\right)\right), \quad B_i=X_i \times\left[0, \sup \beta\left(X_i\right)\right]$$
are measurable in $\mathbb{R}^{n+1}$ with measure
$$m_{n+1}\left(A_i\right)=m_n\left(X_i\right) \cdot \inf \beta\left(X_i\right), \quad m_{n+1}\left(B_i\right)=m_n\left(X_i\right) \cdot \sup \beta\left(X_i\right)$$

## 数学代写|数学分析代写Mathematical Analysis代考|Summable Functions

We would like to extend the notion of integral to real functions that are not necessarily bounded, and defined on measurable sets that may not be bounded.
Throughout this section $X$ will be a measurable domain in $\mathbb{R}^n$ and $f$ a generically continuous function on $X$, meaning continuous on $X$ except for finitely many points at most.

Start from a non-negative function $f: X \rightarrow[0,+\infty)$. Denote by $\mathcal{L}(f)$ the collection of bounded measurable subsets of $X$ on each of which $f$ is bounded and integrable. Then we may define the map
$$F: Y \in \mathcal{L}(f) \rightarrow \int_Y f(x) d x$$

If $X$ is bounded and $f$ is bounded on $X$, the map $F$ in (8.105) is the integral function of $f$. As $f$ is non-negative on $X$,
$$\int_X f(x) d x=\sup \left{\int_Y f(x) d x: Y \in \mathcal{L}(f)\right} .$$
More generally, suppose now $X$ is a measurable set, but possibly unbounded. Let $f$ be a non-negative function which may or not be bounded on $X$. We shall say that $f$ is summable (or integrable in generalised sense) over $X$ if
$$\sup \left{\int_Y f(x) d x: Y \in \mathcal{L}(f)\right}<+\infty .$$
If so, the integral of $f$ over $X$ is, by definition, the above least upper bound. This integral, also referred to as improper integral, is denoted by the usual symbol, and hence equals the real number in (8.106).

Let now $f: X \rightarrow[0,+\infty)$ be a generically continuous function, and suppose $Y_k$ is an increasing sequence of sets in $\mathcal{L}(f)$ such that
$$m\left(X-\bigcup_{k=1}^{\infty} Y_k\right)=0$$

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Properties of Riemann Integrals

$$\alpha(x) \leq \beta(x), \quad \forall x \in X$$

$$D=(x, y) \in X \times \mathbb{R}: \alpha(x) \leq y \leq \beta(x)$$

$$m_{n+1}(D)=\int_X \beta(x)-\alpha(x) d x$$

$$A_i=X_i \times\left[0, \inf \beta\left(X_i\right)\right), \quad B_i=X_i \times\left[0, \sup \beta\left(X_i\right)\right]$$

$$m_{n+1}\left(A_i\right)=m_n\left(X_i\right) \cdot \inf \beta\left(X_i\right), \quad m_{n+1}\left(B_i\right)=m_n\left(X_i\right) \cdot \sup \beta\left(X_i\right)$$

## 数学代写|数学分析代写Mathematical Analysis代考|Summable Functions

$$F: Y \in \mathcal{L}(f) \rightarrow \int_Y f(x) d x$$

$\backslash$ int_ $X f(x) d x=\backslash$ sup $\backslash$ left $\backslash$ int_ $Y f(x) d x: Y \backslash$ in $\backslash$ mathcal ${L}(f) \backslash r i g h t}$ 。

$$m\left(X-\bigcup_{k=1}^{\infty} Y_k\right)=0$$

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