The definition I gave is the standard one, but that’s an interesting question. I’m not sure I know enough measure theory to find a hole in this right away; presumably there is one, because this would be a much more succinct definition.
EDIT: TheMajor pointed me to this SE question; looks like this can work if you can prove that the aforementioned set is measurable in the product sigma-algebra. I like this definition a lot more.
This area definition is equivalent to the standard definition, although this was (to me) not immediately obvious.
Some statements (linearity of integrals, for example) are obvious from the one definition, while others (the Monotone Convergence Theorem) are obvious from the other definition. Unfortunately, proving that the two definitions are equivalent is pretty much the proof for these statements (assuming the other definition).
The general approach of “given a claim, test it on indicator functions, then simple functions, then all integrable positive functions, then all integrable functions, then (if desired) integrable complex functions” is called the standard machine of measure theory, so there is educational benefit to seeing it.
The definition I gave is the standard one, but that’s an interesting question. I’m not sure I know enough measure theory to find a hole in this right away; presumably there is one, because this would be a much more succinct definition.
EDIT: TheMajor pointed me to this SE question; looks like this can work if you can prove that the aforementioned set is measurable in the product sigma-algebra. I like this definition a lot more.
Mathoverflow has discussion on it. In short:
This area definition is equivalent to the standard definition, although this was (to me) not immediately obvious.
Some statements (linearity of integrals, for example) are obvious from the one definition, while others (the Monotone Convergence Theorem) are obvious from the other definition. Unfortunately, proving that the two definitions are equivalent is pretty much the proof for these statements (assuming the other definition).
The general approach of “given a claim, test it on indicator functions, then simple functions, then all integrable positive functions, then all integrable functions, then (if desired) integrable complex functions” is called the standard machine of measure theory, so there is educational benefit to seeing it.