In standard Riemannian integration theory some sort of orientation is defined due to the addition of a minus sign when one switches the limits of integration. In the integration theory on manifolds, an orientation is also provided by the volume form on the manifold.
But in the theory of Lebesgue it doesn't seem to matter because the integration is based on the measure of a set and this is orientation independent. In the literature I found that they do sometimes add some kind of convention when we integrate over intervals (or higher dimensional generalizations) just as in the theory of Riemann. But this feels a bit forced.
Is there any formal way to deduce an orientation based on the properties of the Lebesgue integral? And if not, how does one unite both theories with respect to this concept?