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It is known that if $M$ is an oriented submanifold of $\mathbb{R}^{n}$ and $\omega$ is a compactly supported form on $M$, then \begin{align} \int_{M} \omega = - \int_{-M} \omega, \end{align} where $-M$ denotes $M$ with opposite orientation. Is there an analogous identity for the Lebesgue integral of Lebesgue integrable functions? That is, can one define orientation for arbitrary Lebesgue-measurable sets and their corresponding measures?

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You can of course use the above equation to define an oriented version of the Lebesgue integral, whenever you can define a notion of orientation of the underlying measure space, which is clearly the case for orientable (sub-)manifolds. On the other hand not every measure space admits a meaningful notion of orientation, so the general answer to your question should be no.

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    You actually don't want to define orientation of the measure space itself. Orientation is defined in terms of some other structure of the measure space, for example a vector space structure, or more generally a manifold structure.2011-04-02