Here is my own favorite construction of the (Lebesgue) integral.
Suppose $M$ is an arbitrary smooth manifold. Denote by $Or(M)$ the orientation line bundle of $M$. This bundle is equipped with a canonical Riemannian metric. Vectors of length 1 in the fiber of $Or(M)$ over a point $p∈M$ correspond canonically to the two orientations of the tangent space at the point $p$. The manifold $M$ is orientable if and only if the bundle $Or(M)$ is trivializable. Choosing an orientation of $M$ amounts to choosing an isometric trivialization of $Or(M)$.
The bundle $Or(M)$ together with its natural metric is flat. Hence we can twist the de Rham complex $Ω^0(M)→⋯→Ω^n(M)$ by Or(M) and obtain the following twisted de Rham complex: $Ω^0(M)⊗Or(M)→⋯→Ω^n(M)⊗Or(M)$. (Here by a complex I mean a complex of sheaves.) The line bundle $Ω^n(M)⊗Or(M)$ is called the bundle of densities and is denoted by $Dens(M)$. This bundle has a canonical orientation (hence it is trivializable), but does not have a canonical metric or a canonical trivialization.
The cohomology of the twisted de Rham complex (with compact support) is called the twisted de Rham cohomology (with compact support). We have a canonical map $C^∞_{cs}(Dens(M))→H^n_{cs}(M,Or(M))$. Here $C^∞_{cs}$ is the space of global sections of a vector bundle with compact support and $H^n_{cs}$ denotes the nth cohomology with compact support.
The Poincaré duality gives us a canonical isomorphism $H^n_{cs}(M,Or(M))→H_0(M)$. Finally, the map from $M$ to the point induces a map in homology $H_0(M)→H_0(∙)=R4$.
The composition of maps $C^∞_{cs}(Dens(M))→H^n_{cs}(M,Or(M))→H_0(M)→H_0(∙)=R$ gives us a map $∫: C^∞_{cs}(Dens(M))→R$, which is the integration map. Note that the actual integration (over each connected component) happens in the first map. The second map is an isomorphism and the third map simply sums integrals over individual connected components.
The map $f∈C^∞_cs(Dens(M))→∫|f|∈[0,∞)$ is a norm on $C^∞_{cs}(Dens(M))$. Completing $C^∞_{cs}(Dens(M))$ in this norm yields $L_1(M)(=L^1(M))$, which can be identified with the space of finite complex-valued measures on $M$.
The space of bounded measurable functions on $M (=L_0(M)=L^∞(M))$ can be constructed by completing $C^∞(M)$ in the $σ$-weak topology induced by $L_1(M)$. Other $L_p$ spaces can be constructed in a similar way to $L_1(M)$ by completing sections of the bundle of $p$-densities instead of 1-densities $(=Dens(M))$.
The development of the remainder of measure theory in this approach largely parallels the one explained in one of my previous answers.
I want to stress that these constructions do not rely on any existing integration theory. In fact, they can be used to build integration theory on smooth manifolds from scratch without ever referring to the usual measure theory with its lengthy and technical proofs.
