Recall that a domain of integration in ${\mathbb{R}}^n$ is a bounded subset whose boundary has measure zero.

Def: Let $D\subseteq \mathbb{R}$ be a domain of integration, and let $\omega$ be a continuous $n$-form on $\bar{D}$. Any such form can be written as $\omega =fd{x}^1\wedge \cdots \wedge d{x}^n$ for some continuous function $f:\bar{D}\to \mathbb{R}$. We define the integral of $\omega$ over $D$ to be $$\int_D \omega := \int_D f.$$This can be written more suggestively as $$\int_D f ; dx^1 \wedge \dots \wedge dx^n := \int_D f; dx^1\cdots dx^n.$$
Somewhat more generally, let $U$ be an open subset of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and suppose $\omega$ is compactly supported $n$-form on $U$. We define $$\int_U \omega = \int_D \omega,$$where $D\subseteq {\mathbb{R}}^n$ or ${\mathbb{H}}^n$ is any domain of integration containing $\text{supp }\omega$, and $\omega$ is extended to be zero on the complement of its support.

Prop: Suppose $D$ and $E$ are open domain of integration in ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $G:\bar{D}\to \bar{E}$ is a smooth map that restricts to an orientation-preserving or orientation-reversing diffeomorphism from $D$ to $E$. If $\omega$ is an $n$-form on $\bar{E}$, then $$\int_D G^* \omega = \begin{dcases}\int_E \omega & \text{if }G \text{ is orientation-preserving,} \ -\int_E \omega & \text{if }G\text{ is orientation-reversing.}\end{dcases} $$

Lemma: Suppose $U$ is an open subset of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $K$ is a compact subset of $U$. Then there is a domain of integration $K\subseteq D\subseteq \bar{D}\subseteq U$.

Prop: Suppose $U$, $V$ are open subsets of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $G:U\to V$ is an orientation-preserving or orientation-reversing diffeomorphism. If $\omega$ is a compactly supported $n$-form on $V$, then $$\int_V \omega = \pm \int_U G^*\omega,$$with the positive sign if $G$ is orientation-preserving, and the negative sign otherwise.

Integration on Manifolds

Def: Let $M$ be an oriented smooth $n$-manifold with or without boundary, and let $\omega$ be an $n$-form on $M$. Suppose first that $\omega$ is compactly supported in a domain of a single smooth chart $(U,\phi )$ that is either positively or negatively oriented. We define the integral of $\omega$ over $M$ to be $$\int_M \omega =: \pm \int_{\varphi[U]} \left(\varphi^{-1}\right)^* \omega, $$with the positive sign for positively oriented chart, and the negative sign otherwise.

Prop: With $\omega$ as above, ${\int }_M\omega$ does not depend on the choice of smooth chart whose domain contains $\text{supp }\omega$.

To integrate over an entire manifold, we combine this definition with a partition of unity.

Def: Suppose $M$ is an oriented smooth $n$-manifold with or without boundary, and $\omega$ is a compactly supported $n$-form on $M$. Let $\{U_i\}$ be a finite open cover of $\text{supp }\omega$ by domains of positively or negatively oriented smooth charts and let $\{{\psi }_i\}$ be the subordinates smooth partition of unity. Define the integral of $\omega$ over $M$ to be $$\int_M \omega := \sum_{i} \int_M \psi_i \omega. $$
Prop: The definition of ${\int }_M\omega$ given above doesn't depend on the choice of open cover or partition of unity.

Just we have for orientations, we have a special definition in the zero-dimensional case.

Def: The integral of a compactly supported $0$-form $f$ over an oriented $0$-manifold $M$ is to be defined to be the sum $$\int_M f := \sum_{p\in M}\pm f(p), $$where we take the positive sign where the orientation is positive sign and the negative sign at points where it is negative.

Def: If $S\subseteq M$ is an oriented immersed $k$-dimensional submanifold with or without boundary, and $\omega$ is a $k$-form on $M$ whose restriction to $S$ is compactly supported we interpret ${\int }_S\omega$ to mean ${\int }_S{\iota }_S^{\ast }\omega$, where ${\iota }_S:S\hookrightarrow M$ is inclusion. In particular, if $M$ is a compact, oriented, smooth $n$-manifold with boundary and $\omega$ is an $(n-1)$-form on $M$, we can interpret ${\int }_{\partial M}\omega$ unambiguously as the integral of ${\iota }_{\partial M}^{\ast }\omega$ over $\partial M$, where $\partial M$ is always understood to have the Stoke's orientation.

Properties of Integral of Forms: Suppose $M$ and $N$ are nonempty oriented smooth $n$-manifolds with or without boundary, and $\omega$, $\eta$ are compactly supported $n$-forms on $M$.

• Linearity: If $a,b\in \mathbb{R}$, then $$\int_M a\omega + b\eta = a \int_M \omega + b\int_M \eta. $$
• Orientation Reversal:: If $-M$ denotes $M$ with the opposite orientation, then $$\int_{-M}\omega = -\int_M \omega. $$
• Positivity: If $\omega$ is a positively oriented orientation form, the ${\int }_M\omega >0$.
• Diffeomorphism Invariance: If $F:N\to M$ is an orientation-preserving or orientation-reversing diffeomorphism, then $$\int_M \omega = \begin{dcases}\int_N F^\omega & \text{if }F \text{ is orientation-preserving,} \ -\int_N F^ \omega & \text{if }F\text{ is orientation-reversing.}\end{dcases} $$
  Cor: Suppose $E$ and $M$ are smooth $n$-manifolds with or without boundary, and $\pi :E\to M$ is a smooth $k$-sheeted covering map or generalised covering map. If $E$ and $M$ are oriented and $\pi$ is orientation-preserving $$\int_E \pi^*\omega = k \int_M \omega $$for any compactly supported $n$-form $\omega$ on $M$.

Integrations Over Parametrizations: Let $M$ be an oriented smooth $n$-manifold with or without boundary, and let $\omega$ be a compactly supported $n$-form on $M$. Suppose $D_1,\dots ,D_k$ are open domains of integration in ${\mathbb{R}}^n$, and for $i=1,\dots ,k$, we are given smooth maps $F_i:{\bar{D}}_i\to M$ satisfying:

• $F_i$ restricts to an orientation-preserving diffeomorphism from $D_i$ onto an open subset $W_i\subseteq M$;
• $W_i\cap W_j=\varnothing$ when $i\ne j$;
• $\text{supp }\omega \subseteq \bigcup _{i=1}^k{\bar{W}}_i$.
  Then $$\int_M \omega = \sum_{i = 1}^k \int_{D_i} F^*_i \omega. $$

Integration on Lie Groups

Def: Let $G$ be a Lie group. A covariant tensor field $A$ on $G$ is said to be left-invariant if $L_g^{\ast }A=A$ for all $g\in G$.

Prop: Let $G$ be a compact Lie group endowed with a left-invariant orientation. Then $G$ has a unique positively oriented left-invariant $n$-form ${\omega }_G$ with the property that $$\int_G \omega_G = 1.$$
The orientation form whose existence is asserted in this proposition is called the Haar volume form on $G$. Similarly, the map $f\mapsto {\int }_{G}f{\omega }_G$ is called the Haar integral. This I suspect is just a special case of a Haar Measures

We see that every Lie group has a left-invariant orientation form that is uniquely determined up to constant multiple. It is only in the compact case that we can use the volume normalisation to single out a unique one.