As is well known, the Lebesgue integral of a Lebesgue integrable function on a Lebesgue measure zero set is always zero.
Question: Is the integral of a distribution function(distribution means here the notion in Sobolev space theory or PDE, such as the $L^1$ or $L^2$ or $L^1_{locally}$ functions on domains of $\mathbb{R}^n$ or some kind of manifolds ) on a measure zero set is always zero?
Remark 1 I don't know whether this integral can be called the Lebesgue integral.
Remark2 Are there some kind of measures that the integra value on measure 0 set is not 0?