Given a function $f:\mathbb{R}^2\to\mathbb{R}$, the calculus textbook I'm referencing defines the double integral over a region $D$ by forming a rectangle $R$ that contains $D$, and constructing a function $F(x,y)=\left\{\begin{align*}f(x,y)\ \ \ \text{ for } (x,y)& \in D\\ 0 \ \ \ \ \ \ \ \ \text{ for } (x,y)& \in R\setminus D\end{align*}\right.$ and setting $$\int_D f = \int_R F.$$ The book says this holds as long as $D$ has a "nice enough" boundary with "nice enough" being outside the scope of the book.

I'm wondering what it is, and if it can be stated concisely? Is it some topological condition, like locally euclidean, or oriented? Is it easier to state in $\mathbb{R}^2$ then in general?

  • 1
    $\begingroup$ This is just a guess, but I'd think that the boundary being measure 0 would be nice enough, which non-pathological boundaries will b. $\endgroup$
    – Alan
    Sep 14 at 22:51

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