The concept of multiple integral

4.1 The concept of multiple integral

$D \subseteq R^{n}$ is a bounded closed set, $f : D \to R$ ,

∫ I = f(x)dμ (x ) D

is the multiple integral where $d μ (x)$ is the $n$ -dimension volume infinitesimal at $x$ .

Recall: Riemann integral of 1-dimension.

∫ b ∫ I = a f(x )dx = [a,b]f(x)dx

Definition 4.1.1 Riemann sum. Given a partition $P$ of $[a, b]$ :

a = x < x < ⋅⋅⋅ < x < x = b 0 1 n− 1 n

and a set of symbol points $ξ_{k} \in [x_{k - 1}, x_{k}]$ , then

∑n S(f,P, ξ) = f(ξk)(◟xk-−◝x◜k−1)◞ k=1 = Δxk=|Ik|

$\forall ϵ > 0$ , $\exists δ (ϵ) > 0$ such that $\forall$ a partition $P$ ,

m1≤akx≤n|xk − xk−1| < δ(𝜖) ⇒ ∀ξ = { ξk|ξk ∈ Ik},|S(f,P, ξ) − I| < 𝜖

Then $f$ is Riemann integrable on $[a, b]$ .

Definition 4.1.2 Darboux. Assuming $f : [a, b] \to R$ is bounded¹, given a partition $P$ , let $\begin{aligned} \overset{―}{S} (f, P) & = \sum_{k = 1}^{n} sup_{x \in I_{k}} f (x) | I_{k} | \\ \underset{―}{S} (f, P) & = \sum_{k = 1}^{n} inf_{x \in I_{k}} f (x) | I_{k} | \\ \underset{―}{S} (f, P) & \leq S (f, P, ξ) \leq \overset{―}{S} (f, P) \end{aligned}$

If $P$ is more dense, then $\overset{―}{S} (f, P)$ decreases and $\underset{―}{S} (f, P)$ increases. If $\forall ϵ > 0$ , $\exists$ a partition $P$ such that

I − 𝜖 < S (f,P ) ≤ I ≤ S(f,P ) < I + 𝜖 ⇔ S-(f,P ) − S (f,P ) < 2𝜖 -- --

Then $f$ is Darboux integrable on $[a, b]$ .

Theorem 4.1.3 The following 3 statements are equivalent.

1.: (Riemann) $f$ is Riemann integrable on $[a, b]$ .
2.: (Darboux) $f$ is Darboux integrable on $[a, b]$ .
3.: (Lebesgue) $f$ is bounded, and ${x \in [a, b] | x is the discontinuity of f}$ is a set of measure $0$ .

Definition 4.1.4 $A \subseteq R$ is a set of measure $0$ if $\forall ϵ > 0$ , $\exists$ countable intervals $I_{1}, . . ., I_{n}, . . .$ such that $A \subseteq I_{1} \cup \dots \cup I_{n} \cup \dots$ and $\sum_{k \geq 1} | I_{k} | < ϵ$ .

Corollary 4.1.5 All continuous functions on $[a, b]$ are integrable. All monotonous functions on $[a, b]$ are integrable.

Rectangle in $R^{n}$ .

Definition 4.1.6 Assuming

n R = [a1,b1] × [an,bn] = {x = (x1,...,xn ) ∈ ℝ |ai ≤ xi ≤ bi,i = 1,...,n}

then $u (R) = (b_{1} - a_{1}) \cdot \dots \cdot (b_{n} - a_{n})$ is called the $n$ -dimension volume of $R$ .

Definition 4.1.7 $R_{1}, . . ., R_{N}$ are the partition $P$ of the rectangle $R$ if

$R = R_{1} \cup \dots \cup R_{N}$ where $R_{k}$ are all rectangles.
$R_{i} \cap R_{j} \subseteq \partial R_{i} \cap \partial R_{j}$ , $\forall i, j$ .

Definition 4.1.8 Riemann sum in higher dimension.

N ∑ S (f,P,ξ) = f(ξk)◟|Rk◝◜|◞ k=1 μ(Rk)

$f$ is Riemann integrable on rectangle $R$ if $\exists I \in R$ such that $\forall ϵ > 0$ , $\exists δ (ϵ) > 0$ such that for any partition $P = {R_{k} | 1 \leq k \leq N}$ of $R$ ,

max sup ∥x − y ∥ < δ(𝜖) ⇒ ∀ξ = { ξk|ξk ∈ Rk },|S (f,P,ξ ) − I| < 𝜖 1≤k≤nx,y∈Rk

Mark

∫ I = f(x)dμ(x ) R

as the Riemann integral of $f$ on $R$ . When the dimension $n > 1$ , it’s also called the multiple integral.

Definition 4.1.9 Darboux.

-- ∑N ∑N S(f,P ) = sup f(x )μ (Rk), S(f,P ) = inf f(x )μ (Rk) k=1 x∈Rk k=1 x∈Rk

$f$ is Darboux integral on rectangle $R$ if $\dots$

Theorem 4.1.10 The following 3 statements are equivalent for any function defined on rectangle $f : R \to R$ .

1.: (Riemann) $f$ is Riemann integrable on $[a, b]$ .
2.: (Darboux) $f$ is Darboux integrable on $[a, b]$ .
3.: (Lebesgue) $f$ is bounded, and ${x \in R | x is the discontinuity of f}$ is a set of measure $0$ .

Corollary 4.1.11 All continuous functions on bounded closed rectangle $R$ are integrable.

Assuming $D$ is a bounded closed set of any shape, $f : D \to R$ , how to seek $\int_{D} f (x) d μ (x)$ ?

Firstly, take a rectangle $R \in R^{n}$ such that $D \subseteq R$ .
Construct
$fR =$
If $f_{R}$ is integrable on $R$ , then $f$ is called integrable on $D$ , and define
$∫ ∫ f(x)d μ(x) = fR(x )d μ(x) D R$
It requires that $\partial D$ is a set of measure $0$ .

Thus, we have

Definition 4.1.12 Assuming $D$ is a bounded closed set where $\partial D$ is a set of measure $0$ , $f : D \to R$ . Let rectangle $R \in R^{n}$ satisfy $D \subseteq R$ , construct

fR =

If $f_{R}$ is integrable on $R$ , then $f$ is integrable on $D$ , and

∫ ∫ f(x)d μ(x) = f (x )d μ(x) D R R

Corollary 4.1.13 If $D$ is bounded closed where $\partial D$ is a set of measure $0$ , then all continuous functions on $D$ is integrable.

Corollary 4.1.14 Indicator function

1D (x ) =

is Riemann integrable on $D$ , define $μ (D) = \int_{D} 1 \cdot d μ (x)$ .

Properties. Let

R (D) = {f : D \to R is Riemann integrable}

C (D) ⊊ R (D)

1.

Linearity. $R (D)$ is a linear space, $f \mapsto \int_{D} f d μ$ is linear, i.e. $\forall f, g \in R (D)$ , $\forall α, β \in R$ , $α f + β g \in R (D)$ and

∫ ∫ ∫ (αf + βg)d μ = α fdμ + β gd μ D D D

2.

Isotonicity. $f, g \in R (D)$ and $\forall x, f (x) \leq g (x)$ , then

∫ ∫ f(x)dμ ≤ g(x)dμ D D

3.

Triangle inequality. $f \in R (D) \Leftrightarrow | f | \in R (D)$ , and $- | f | \leq f \leq | f |$ , so

∫ ∫ ∖left| f (x)dμ∖right | ≤ |f(x )|dμ D D

4.

Cauchy-Schwartz inequality. $f, g \in R (D) \Rightarrow f g \in R (D)$ , then

--------------------- ∫ ∘ ∫ ∫ ∖lef t| f(x)g(x)dμ ∖right| ≤ f(x )dμ g(x)dμ D D D

5.

Integral mean theorem. Assuming $D$ is (path-)connected, $g \in R (D)$ and $g (x) \geq 0$ , $f \in C (D) \subseteq R (D)$ , then $\exists ξ \in D$ , such that

∫ ∫ f(x)g(x)d μ = f(ξ) g(X )dμ D D

When the denominator is non-zero, we have

∫ f (x)g(x)dμ f(ξ) = -D∫------------ D g(x)dμ

It is a weighted mean of $f$ where $g$ is the weight.

¹It could be proved from the definition of Riemann sum that if $f$ is Riemann integrable, then $f$ is bounded on $[a, b]$ .