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I'm trying to understand this paper, and I see this integral (page 2):

$$ \int_{B\ \cap\ \mathcal{C}} (1 - y)dy, $$ where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any measurable set in $[0, 1]^2$.

This seems like a line integral, but the definition of a line integral that I know requires a differentiable parametrization of the curve. But

  • The direction of the curve is not specified, so should the integral be positive or negative? (I know it should be positive, but what's the mathematical reason?)
  • What if $B$ is such that $B \cap \mathcal{C}$ is not connected? No parametrization from an interval to $B \cap \mathcal{C}$ could be continuous, let alone differentiable.

I though of simply projecting $B \cap \mathcal{C}$ onto the $y$ axis and taking a one dimensional Lebesgue integral over the projection, but according to this question the projected set may not be measurable.

So what is the definition of an integral like this?

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    $\begingroup$ Without specified orientation on the curve, this integral is undefined. $\endgroup$
    – Moishe Kohan
    Commented Jul 1 at 0:38
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    $\begingroup$ Presumably they mean the measure $|dy|$. Who knows! $\endgroup$
    – Ted Shifrin
    Commented Jul 1 at 1:03

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