I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example,
$$ ∫_{[a,b]}fdm=-∫_{[b,a]}fdm $$
If b>a, I know that the right hand side is nonsense but.. I would like to know some similar results on it. I believe that there may have been some consideration on it among mathematicians. In fact, there are some comment in wiki as in the below but I don't know what it means. Any help would be appreciate.
http://en.wikipedia.org/wiki/Lebesgue_integration
Domain of integration[edit] A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation:
$$ ∫_b^a f := - ∫_a^b f. $$
Generalizing this to higher dimensions yields integration of differential forms. By contrast, Lebesgue integration provides an alternative generalization, integrating over subsets with respect to a measure; this can be notated as
$$ ∫_A f\,d\mu = ∫_{[a,b]} f\,d\mu $$
to indicate integration over a subset A. For details on the relation between these generalizations, see Differential form: Relation with measures.
If $\mu$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, then we usually define $$ \int_a^b f\,\mathrm d\mu:=\int_{(a,b]}f\,\mathrm d\mu=\int_\mathbb{R}f\mathbf{1}_{(a,b]}\,\mathrm d\mu $$ for integrable $f$ and $a<b$. In this case we have $$ \int_a^c f\,\mathrm d\mu=\int_a^b f\,\mathrm d\mu +\int_b^c f\,\mathrm d\mu\tag{1} $$ for $a<b<c$ since $\mathbf{1}_{(a,c]}=\mathbf{1}_{(a,b]}+\mathbf{1}_{(b,c]}$. If we use the convention that $$ \int_a^b f\,\mathrm d\mu=-\int_b^c f\,\mathrm d\mu $$ whenever $b<a$, then $(1)$ holds for any choice of $a,b,c\in\mathbb{R}$.