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167 Post author: lukeprog 16 January 2011 08:30AM

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Comment deleted 17 January 2011 04:28:14PM *  [-]
Comment author: [deleted] 17 January 2011 05:04:38PM *  6 points [-]

Hmm. Upvoted for contributing to a good topic but I'm not sure I agree.

I just looked up the gauge integral because I wasn't familiar with it. For those curious about the debate, here's the introduction to the gauge integral I found, which has a lot of relevant information. My beef with this is precisely that it doesn't use the general background of measure theory (sigma-algebras, measurable functions, etc.) and you're going to need that background to do useful things. The gauge integral approach doesn't give you the tools to generalize to scenarios like Brownian motion where you need to construct different measures; also, the gauge integral doesn't come with a lot of nice convergence theorems the way the Lebesgue measure does.

I don't find the standard treatment of measure theory especially hard; it takes about a month to understand everything up to the Lebesgue integral, which isn't an obscene time commitment.

Also, there's some virtue to just being familiar with the definitions and concepts that everybody else is. (It's not just mathematicians "refusing to update." I know for sure that economists, and potentially people in other fields, speak the language of standard measure theory. But maybe it's not everyone. What are you using measure theory for?)

If you're looking for an easier, more straightforward treatment than Rudin, I'd recommend Cohn's Measure Theory. I'm not sure why, but it feels friendlier and less digressive.

Comment deleted 18 January 2011 06:14:40AM [-]
Comment author: komponisto 18 January 2011 07:50:54AM 1 point [-]

I also rank Halmos higher than Cohn in terms of measure theory books

Halmos, by the way, is a top-notch mathematical author in general. Every one of his books is excellent. Finite-Dimensional Vector Spaces in particular is a classic.

Comment author: rich 18 January 2011 12:06:22PM 0 points [-]

I think that's the only book I kept from my Maths degree. In hardback, too. I have lent it to a colleague and keep a careful eye on where it is every couple of weeks...

Comment author: JoshuaZ 17 January 2011 05:12:14PM 0 points [-]

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.

Comment author: Will_Sawin 17 January 2011 08:53:46PM 3 points [-]

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.

Comment deleted 18 January 2011 06:23:02AM [-]
Comment author: JoshuaZ 18 January 2011 02:48:14PM 0 points [-]

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.

Comment author: jsalvatier 17 January 2011 04:55:59PM *  0 points [-]

I haven't studied real analysis, could you explain what advantages the guage integral is better than the lebesgue integral? Edit: maybe just respond to SarahC.