I’m not sure why you consider the gauge integral to be easier to understand the Lebesgue integral. It may be due to learning Lebesgue first, but I find it much more intuitive.
Also:
Royden is slightly better in this respect. The first four chapters are excellent, but still probably too theoretical. Further, eventually one will encounter measure spaces that aren’t based on the real numbers and the Lebesgue measure,
Yes, this is a good thing. One doesn’t understand a structure until one understands which parts of a structure are forcing which properties. Moreover, this supplies useful counterexamples that helps one understand what sort of things one necessarily will need to invoke if one wants certain results.
I’m not sure you made it to the end of that sentence. It is a good thing, but Royden obscures the connection with this four-chapter long digression into metric spaces and Banach spaces.
Maybe I misunderstand the needs of a budding Bayesian analyst. I agree that these sorts of counterexamples are useful, but perhaps not up front. I find it hard to imagine they’d encounter measure spaces that are either 1) not discrete or 2) not a subspace of R^n on a regular basis.
I’m not sure why you consider the gauge integral to be easier to understand the Lebesgue integral. It may be due to learning Lebesgue first, but I find it much more intuitive.
Also:
Yes, this is a good thing. One doesn’t understand a structure until one understands which parts of a structure are forcing which properties. Moreover, this supplies useful counterexamples that helps one understand what sort of things one necessarily will need to invoke if one wants certain results.
It is much easier to understand what the words in the definition of the gauge integral mean. It is harder to understand why they are there.
I’m not sure you made it to the end of that sentence. It is a good thing, but Royden obscures the connection with this four-chapter long digression into metric spaces and Banach spaces.
Maybe I misunderstand the needs of a budding Bayesian analyst. I agree that these sorts of counterexamples are useful, but perhaps not up front. I find it hard to imagine they’d encounter measure spaces that are either 1) not discrete or 2) not a subspace of R^n on a regular basis.
Yes, I did make it to the end of the sentence. But I misinterpreted the sentence to be having two distinct criticisms when there was only one.