Sorry if this is kind of vague, but the other day, one of my math profs told me about a theorem which he thought was particularly interesting. I might be missing or getting a condition wrong, but here goes:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose [tex]I(f, d)[/tex] is a real-valued function, where f is a real-valued function always defined on a region d, and that region d. if:

1. [tex]I(f_1 + f_2, d) = I(f_1, d) + I(f_2, d)[/tex]

2. If [tex]d = d_1 \cup d_2[/tex], and [tex]d_1 \cap d_2 = \emptyset[/tex], then [tex]I(f, d) = I(f, d_1) + I(f, d_2)[/tex]

3. If [itex]a[/itex] is a real constant, then [tex]I(af, d) = aI(f, d)[/tex]

Then, [tex]I(f, d) = \int_d f[/tex]

I asked about where I could find this in a book or online and he said a graduate-level book on measure theory (which unfortunately I don't have any). I did look through Spivak's calculus on manifolds, but nothing came close even.

Does anyone recognize this?

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# Interesting math theorem in measure theory

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