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(∙)=R.

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 the bundle of 1-densities Dens(M).

The development of the remainder of measure theory in this approach largely parallels
the one explained in <a href="https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820">one of my previous answers</a>.

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.

**Added later:** In the middle of the proof, we relied on Poincaré duality, which can be most easily established using sheaf cohomology.
To this end, one must show that the de Rham cohomology is isomorphic to the sheaf cohomology with real coefficients.
This boils down immediately to the Poincaré lemma.

The simplest way to establish the Poincaré lemma is as follows.
The de Rham complex of a finite-dimensional smooth manifold M
is the free C^∞-dg-ring on the C^∞-ring C^∞(M).
If M is the underlying smooth manifold of a finite-dimensional
real vector space V, then C^∞(M) is the free C^∞-ring on
the vector space V* (the real dual of V).
Thus, the de Rham complex of a finite-dimensional real vector space V
is the free C^∞-dg-ring on the vector space V*.
This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain
complex on the vector space V*.
The latter cochain complex is simply V*→V* with the identity differential.
It is cochain homotopy equivalent to the zero cochain complex,
and the free functor from cochain complexes to C^∞-dg-rings
preserves cochain homotopy equivalences.
Thus, the de Rham complex of the smooth manifold V
is cochain homotopy equivalent to the free C^∞-dg-ring on
the zero cochain complex, i.e., **R** in degree 0.