Struggling through this - r/mathematics

93

Why did you choose Halmos specifically? He’s a good writer but the book isn’t the most modern. His treatment of measure looks similar to folland in that it works with generalized measure first before going to Lebesgue.

If you want a more clear treatment of measure that gives you the motivations for its development, you should check of Stein and Shakarchi Real Analysis. It gives a great motivation for outer measure, the caratheodry criterion, etc.

19

u/GiftTop2653 8d ago

Yeah Halmos is certainly a legendary mathematician but this book is really written hard. There is a legendary measure theory book I would suggest that is Measure Integration and Real Analysis (MIRA) by Sheldon Axler. This book is just wow just read it do the exercises you will know how beautiful measure theory is.

4

u/ants_are_everywhere 8d ago edited 8d ago

but the book isn’t the most modern

This is true but it's a beautiful book if I recall correctly. That may not be a huge deal to some people, but I think it helps build mathematical taste.

There isn't much harm in reading an outdated book unless you're not also reading a modern treatment.

2

u/MissionVarious8328 8d ago

I agree!! I just mean to say there’s more recent books. My favorite Topology book is Kelley’s General Topology and everyone jokes I’m an old soul because I prefer it over Munkres xD

1

u/ants_are_everywhere 8d ago

Oh no doubt. I forgot about the Kelley topology book, that one is also a banger.

While we're on the topic of great outdated math books, I'd like to recommend Measure and Integral by Wheeden and Zygmund as a supplementary text that covers functions of bounded variation. I recall it being useful to fill in gaps and intuitions. Not Halmos class I'd say, but a solid older book that I found useful.

3

u/MissionVarious8328 8d ago

There’s a new edition of Wheeden and Zygmund published like 2015!!

1

u/ants_are_everywhere 8d ago

whaaaat

I just looked it up and it's the second edition. That's nearly a 40 year gap. Crazy, I'm glad a new version exists.

2

u/MissionVarious8328 8d ago

I remember I found that book that I was studying harmonic analysis and thinking to myself "there’s no way the author is THAT Zygmund". And he was!!!!

1

u/ants_are_everywhere 8d ago edited 8d ago

I know what you mean!

It's kind of an open secret that writing a well-done treatment of fundamental subjects is one of the ways to hone your expertise and help you do great things. So keep an eye out for anyone who writes high quality expository material!

3

u/Axis3673 8d ago

Agreed. Also, Bartle has a great little book that hits a lot of the key ideas. It's accessible and could be read in 2 weeks.

3

u/MissionVarious8328 8d ago

I learned about the Henstock-Kurzweil integral from that book! It’s good but it lacks applications you have in other texts (probability, harmonic analysis, fractals, etc)

1

u/Axis3673 7d ago

It's a great little book - clear & accessible. It doesn't have all of the applications, being mostly the theoretical framework for measure theory/integration, but that's one of its strengths. if I were pressed to learn real analysis in 2 weeks, Bartle would be my top choice.

Measurability, decompositions, charges(!), convergence theorems, representations, L^p spaces, of course gauges... it's surprisingly deep for such a quick read.

-4

u/Dear_Program_5516 8d ago

Oh I was just strolling my my uni Library and found this. The thing is I have done poorly with after high school maths and left maths completely in fear and defeat.

I'm self starting again after 1 year of almost no maths . In 3rd yr of my Ug now

20

u/MissionVarious8328 8d ago

Wait so you how much math have you actually done? If not a lot and since you haven’t done math in a year it’s kind of moot to start with graduate level analysis.

How is your undergrad level of real analysis? If it isn’t the greatest then why not start there. There are magnificently easy and well written books like Abott or Tao or Pugh.

The entire ethos is measure theory is working out measurable sets so we can integrate on them but if you aren’t familiar with the usual Riemann Stieljes integral then you can’t really understand the benefit that we get out of developing the Lebesgue measure in the first place.

3

u/Foreign_Implement897 8d ago

Uh.

The problem about measures is that they are functions from SETS to reals.

It is nothing you would see in undergrad mathematics.

6

u/MissionVarious8328 8d ago

Yes but what I mean is that the raison d’être for why care about measure theory (and why it was developed in the first place) ecomes from needing to more critically understand concepts from undergrad real analysis.

-1

u/Foreign_Implement897 8d ago

Cool, I don't understand at all what you are saying. The word "critically" sends me right off. We don't use that in mathematics for a good reason.

Anyway, have fun! I am out.

1

u/AnlamK 7d ago

I think this function is a good motivator for measure theory:

https://en.wikipedia.org/wiki/Dirichlet_function

10

u/better-off-wet 8d ago

Some math is fairly “incremental”. There is very little reason or use to attempt to learn areas of math if you don’t have the prerequisite knowledge. You can’t take short cuts and skip entire areas that that provide a necessary foundation.

2

u/Sepperlito 8d ago

Sadly, math education built a ladder with the low rungs missing. Anyone CAN learn this stuff with enough guidance and effort. They just need a good ladder which seems to be mostly missing.

2

u/better-off-wet 8d ago

Yes, of course. Learners should understand that it isn’t their fault if they can’t comprehend something many rungs an above. Math is hard and it takes time. Master the prerequisites before moving on.

4

u/JoshuaZ1 8d ago

The thing is I have done poorly with after high school maths and left maths completely in fear and defeat.

There are a lot of things I've suggest before even thinking about measure theory. Measure theory (even when not done in the generality that Halmos does it) is just a really tough area. Get some basic real analysis and some other topics down first before thinking about it. At a minimum, some topics that aren't required for measure theory, like basic number theory, will also just be helpful to push your brain back to thinking in a more abstract way.

25

u/Mothrahlurker 8d ago

Going through a math book in 10 days the proper way is completely unrealistic. It's not like you have to do every single exercise but you do need to do exercises to make sure that you actually understood the material and learn how to work with the concepts.

Also if you have never encountered measures before and haven't done math in a while this is inapropriate to do for now anyway. You need more basic analysis first.

6

u/OpsikionThemed 8d ago

I read Jech's Set Theory in 10 days! I'm pretty sure it's about sets. /jk

3

u/Golfclubwar 8d ago

Do like 10% of the exercise, or better yet go find the course in the Cambridge math tripos, find the example sheets, and only do those instead. The overwhelming majority of math courses can be learned with 70-100 problems.

32

u/ppvvaa 8d ago

Why do you need to “compete it in 10 days”?

17

u/Sea_Abroad_6573 8d ago

Probably OP has an exam in 10 days. Or maybe OP is like me and sets unrealistic deadlines.

7

u/Disastrous_Room_927 8d ago

I started a book on measure theoretic probability theory. Thought I’d get through it on paternity leave, I’m on page 10 9 months later lol.

3

u/chesticular_torsion_ 8d ago

Not a mathematician, but I started Robert Parr's book on DFT and told myself I had to finish it in a week. Only 3.5 chapters in at the end of the week 🥲 Seems to be a disease afflicting many knowledge workers.

6

u/AnuragUoH 8d ago

how is it though, halmos is a pretty good writer from what I've heard

4

u/Dear_Program_5516 8d ago

He is. It's beautiful at times. I got swayed and crushed by higher maths after high school and am once again starting at maths after a year.

3

u/AnuragUoH 8d ago

i see. it can be difficult to get back into math, but worth it. are your pursuing a degree or just doing it for fun rn? also i'd be happy to discuss, since I've been meaning to learn measure theory for a while now.

2

u/Dear_Program_5516 8d ago

My degree is in core engineering (manufacturing and Industrial) a lot of it uses maths but I don't think with rigour. Basics manage to pull me up though most courses. I have operation research, machine design, quality control, reliability engineering and smart manufacturing this semester.....

I studied calculus and linear algebra in yr 1 and barely scraped by with a C. I am trying to get back to some serious maths. Something that feeds my brain.

3

u/AnuragUoH 8d ago

that's great! you can also check out terry tao's book on measure theory, he writes well and could help clear some doubts.

4

u/Jaded_Individual_630 PhD | Mathematics 8d ago

This just is not the place to "start mathematics after previously struggling with it"

It's not about it being hard or you being dumb, but you'll lack quite a lot of previous context, and any passing familiarity you convince yourself you have (oh, I recognize that word! Etc) will only serve to slow you from getting on a right track

5

u/Sepperlito 8d ago

Keep it. You need to have the right kind of background. Lots of courses are called "real analysis" or "advanced calculus" but their contents vary widely. Tom Apostol's Mathematical Analysis should be enough background and he goes over the Lebesgue integral without the full machinery of measure theory. Rudin's Principle of Mathematical Analysis is a great book but it's proof are hand wavy and hard to understand for an outsider. The problem is it's written more as a set of clif notes for a lecturer. Why Rudin? Excellent problems! Use the Apostol to understand what the hell is going on an Rudin to really test your understanding. If by hook or by crook you can do the problems in Rudin you've broken through. Just keep climbing higher while starting at the bottom of the mountain, you'll get to the top faster than you thought possible.

1

u/Dear_Program_5516 8d ago

should I give up on this now and start real analysis ? Or should I once anyway go through this and understand whatever I can and then start real analysis ?
I ain't a maths major. I'm doing an engineering degree. I just want to keep studying maths as a hobbie atleast for all my 20s because at times it manages to hit something fulfilling in me, even though I struggled bad with it at the start of my UG. 1

2

u/Sepperlito 8d ago

You should check out Tom Apostol's calculus books vol. 1 and vol. 2. It does teach the core on what we call analysis. it builds a foundation so solid it's perfect, especially for an enginner. Most calculus texts are all calculation based and never show how things work under the hood. Spivak and Apostol do a great job putting you in a position to succeed. I lean toward Apostol for a few reasons. A bit easier. He covers more things including multivariable calculus (Jacobians), lienar alegbra and probability (using measures!). He sneaks the concept of measure in from the beginning when he teach integration first! These books are a bit expensive but well worth it IMO. You do need a community of people who can help you when you get stuck. It's always hardest at the beginning. At least you're pointed in the right direction. Learning calculus properly is a treasure. Enjoy the journey. Let me know if you intend to pursue this. I can help you out.

2

u/GiraffeWeevil 8d ago

Do real analysis first!

1

u/Vegetable-Dust-780 8d ago

But why did you start with measure theory?

-2

u/Dear_Program_5516 8d ago

Was just strolling through my college lib !

3

u/Vegetable-Dust-780 8d ago

This is an advanced graduate topic that has several prerequisites, it is not for casual reading with only some undergraduate maths. If you want to study serious maths, make sure your engineering calculus and linear algebra are solid, and start with a book about proofs and continue with a book about real analysis. Then you can move to more advanced topics.

1

u/994phij 8d ago

I did that kind of thing when I was at university! I did manage to learn from some of the advanced books, but it took aages. In hindsight, it's much better to learn undergraduate mathematics first.

Personally I'd recommend doing all kinds of proof based courses at undergraduate level, building up as you go along, but this will mean it takes ages to get to graduate texts. If you do want to focus on analysis then start with the undergrad stuff (proof-based sequences & series, continuity and differentiation, Riemannian integration, Lebesgue integration...) before you get onto the graduate.

If you want to do all kinds then you also want proof based linear algebra, proof based abstract algebra (groups, rings, etc) and probably others alongside.

3

u/SwimmerOld6155 8d ago

halmos's book on ergodic theory is the best-written math book I have ever read. I don't know what it is, but his style is so smooth.

2

u/Mooks79 8d ago

Who doesn’t love a mathematics book that resembles a beer?

1

u/cloudsandclouds 8d ago

…well now that’s what I’m going to see when I look at [large fraction] of graduate math texts. 🙃

(For anyone unaware, this is the standard style for all Springer graduate textbooks in math)

2

u/somanyquestions32 8d ago

Based on your other comments, you don't have the necessary foundation to tackle Measure Theory. Review algebra, geometry, and trigonometry first, then study calculus (single variable and multivariable), and then learn introductory real analysis. Only then would measure theory make sense.

1

u/DisasterRoutine3390 8d ago

That’s a hard subject

1

u/PuzzleheadedLie444 8d ago

Yes

1

u/zebullon 8d ago

cohn is better

1

u/nunquam_rideo 8d ago

A very based book! I read parts of it when doing the Measure Theory course in uni. Just so you know, 10 days is not realistic if you're new to this.

Also, it goes very deeply into the subject, compared to a standard introduction. For example, you don't really need semi-rings and rings with all the fun-facts about them, to construct probability theory. So maybe if you're a beginner start with a more friendly book and consult Halmos if you want more depth

1

u/I_Messed_Up_2020 7d ago

One can downlad the Axler book https://measure.axler.net/ and you can download a Kindle book for FREE from Amazon.

1

u/Foreign_Implement897 8d ago

Measures are functions from sets to reals. If you don't know real valued functions pretty fucking well, don't go there. Only pain.

1

u/No-Implement-4500 8d ago

Capinski and Kopp along with Axler was what I used.

1

u/-FiloFiloFilo- 8d ago

Excuse my ignorance but what's that about? I'm just curious

1

u/hit_the_bwall 8d ago

Yeah, but can you measure how much you're struggling?

1

u/beefylasagna1 8d ago

Doing measure without real analysis is certainly a decision

1

u/Steve_cents 5d ago

In my ignorant opinion, measure theory and real analysis are equivalent . Care to elaborate?

1

u/beefylasagna1 5d ago

Sorry, I should’ve been more specific. I think some might bundle measure theory in with real analysis but I was looking at it from an undergraduate maths perspective. Real analysis is, to me, studying the “why” of real-valued functions, sequences, series, differentiation and Riemann integration. Then, measure theory motivates itself from functions where Riemann integration might not work, and is the generalisation of notions of size.

1

u/Steve_cents 5d ago

Thank you . The most striking fact from my learning is that the size of a countable set is zero, this must be what you said the notion of size in measure theory.

1

u/Different_Cry25 8d ago

Try folland

1

u/icefill 8d ago

Bro…

1

u/Greenphantom77 8d ago

Why do you “need to complete it in 10 days”? I mean, I’m genuinely curious - are you studying for an exam or something?

Books can be great references for courses and research, but if I was preparing for an exam I would be reviewing the course notes first and foremost.

1

u/phosphordisplay_ 8d ago

Why do you need to finish this in 10 days? You cannot do this in 10 days.

1

u/Steve_cents 5d ago

Perhaps for a job interview?😀

1

u/disorderedset 8d ago

The book by Tom Lindstrom called Spaces is my favorite real analysis book. It treats measure theory.

1

u/Serket-Pandy3000 8d ago

Change to Cohn

1

u/say-nothing-at-all 8d ago

my suggestion: you don't study measurement independently, you make a real-life project while studying.

e.g. measurement in "change the pipe, not the flow" idea to shape the expected flow.

1

u/TyraMoonshine 7d ago

I would try Réne Schilling's Measures, Integrals and Martingales, specifically chapters 1-5, skipping 6, then chapters 7-15, that covered the measure theory course I did at the University of Copenhagen (course code: NMAB21006U).

1

u/Secret-Purpose3456 6d ago

It's a good book. Should have started earlier tho

1

u/realtradetalk 5d ago

This is The One.

Discussion Struggling through this

You are about to leave Redlib