This seminar by somiaj took place on 27th July 2008 20:00 UTC, in #mathematics channel of

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Topic Edit

construction of the Lebesgue Measure

Seminar Edit

21:00:00 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS.
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20:59:20 ~Mirchiss: somiaj: what do you need to know before hand for the lecture?
21:00:42 somiaj: Mirchiss: it is again going to be mostly definitions and ideas, a bit of
                 real analysis is needed but not much more beyond that.

21:01:58 somiaj: so anyways last time we talked about the abstract structure. For a measure
                 space we have a set Omega, from Omega we have a collection of subests of
                 Omega called Sigma that is a sigma-field (closed under countable unions,
                 intersections and complements)
21:03:04 somiaj: finally we had a function mu: Sigma -> [0,infinity] in which was
                 countabally additive (basically meaning the volume of an object is the sum
                 of the volume of its disjoint pieces)
21:03:33 somiaj: so today we are going to talk about creating a measure, in paticularlly the
                 lebesgue measure.
21:04:26 somiaj: The lebesgue measure is created from what is known as an outer measure. If
                 P(Omega) is the power set on omega, then we define
                 mu* : P(Omega) -> [0,infity] to be an outer measure if
21:04:44 somiaj: 1) mu*(emptyset) = 0
21:04:59 somiaj: 2) mu*(A) <= mu*(B) whenever A subset B
21:05:25 somiaj: 3) mu* is countably sub-additive, mu*( union_n A_n ) <= sum_n mu*(A_n)
21:06:08 somiaj: so basically an outter measure is a function defeind on the whole powerset
                 that has some of the desired properties.
21:06:23 somiaj: The bigest example of an outer measure if the lebesgue outter measure.
21:07:47 somiaj: To define this outmeausre lets first define an n-cube. an n-cube is a
                 subset of R^n of the form prod_{k=1}^n [a_k,b_k]
21:08:37 somiaj: where a_k < b_k are real numbers. in R^1 this is just an interval, in R^2
                 this is a rectangle, in R^3 it is a 'cube' (can't really think of another
                 word but don't think cube means all sides are the same size)
21:09:09 somiaj: and prod_{k=1}^n is the cartiseian product of intervals.
21:10:25 somiaj: We can then define the volume of an n-cube in the standard way. if
                 I = prod_{k=1}^n [a_k,b_k] then the volume of I is,
                 v(I) = prod_{k=1}^n (b_k - a_k) [here we are using the usual product on
                 the real numbers)
21:10:43 somiaj: again, in R^1 this gives us length, in R^2 this gives us area, R^3 volume
                 and so forth.
21:11:16 somiaj: From this we can define the outer measure for any A subest R^n. We define
                 this outer measure to be
21:12:21 somiaj: mu*(A) = inf_{A subest union I_k} sum_k v(I_k) where
                  {I_k}_{k a natural number} is any cover of n-cubes of the subest A.
21:13:16 somiaj: (note, I guess to alivate confusion better to call these n-rectangles
                  instead of cubes)
21:15:06 Jafet: !
21:16:02 somiaj: So we need to show that this creation is an outer measure. First off it is
                 well defined. for any cover of n-rectangles, sum_k v(I_k) is a real number
                 in the range [0,infinity]. And if you have a bit of real analysis any
                 non-negative collection of real numbers has an infimum that is also in the
                 range [0,infinity]
21:16:07 somiaj: yes Jafet
21:16:14 Jafet: somiaj, that is, mu*(A) is the volume of some arbitrary bunch of n-cubes
                that happen to contain all the elements in A?
21:16:48 somiaj: Jafet: mu*(A) is the infimum of all possible covers of the subest A.
21:17:22 ichor: !
21:18:29 somiaj: ichor: yes.
21:19:06 ichor: What is a cover of n-cubes?
21:20:01 somiaj: ichor: {I_k} is a cover of the set A, if A subest union_k I_k
21:20:15 kommodore: I think you want _by_ n-rectangles
21:20:35 somiaj: yea, lets use n-rectangles instead, I think of them as n-cubes, but that
                 is confusing to normal termology.
21:20:35 ichor: somiaj: Ah, I see. Ok.
21:20:48 Jafet: somiaj, "box"?
21:21:03 somiaj: Jafet: that could work as well.
21:21:46 somiaj: So lets say you are given an arbitary set in R^2, to find its area you find
                 the area of countable collection of n-rectangles that cover the set A
21:22:32 somiaj: This is because we can calculate the area of this collection of
                 n-rectangles. We then take the infimum over all possible covers.
21:23:52 somiaj: As I said this forms an outer measure. First off every collection of
                 extended real numbers [0,infinity] has an infimum that is also a real
21:24:27 somiaj: So we only need to show that it satisfies the three properties I listed for
                 an outermeasure.
21:25:05 somiaj: mu*(emptyset) = 0 is straight forward, consider an empty cover, the volume
                 of the empty cover is 0, thus mu*(emptyset) = 0.
21:25:40 somiaj: second if A subset B, then every cover {I_k} of B is also a cover of A,
                 thus it follows that mu*(A) <= mu*(B)
21:26:39 somiaj: the third is the hardest to show. Suppose {A_n} is a collection of subsets,
                 I need to show that mu*( union_n A_n ) <= sum_n mu*(A_n)
21:27:14 somiaj: this inequality is clear if mu*(A_n) = infinity for any n, so we can assume
                 that mu*(A_n) < infinity for all n.
21:28:09 somiaj: Next we find an 'efficient cover' for each A_n. Give e>0 there exists a
                 cover {I_nk}_{k in N} of A_n such that
21:28:45 somiaj: mu*(A_n) <= sum_k v(I_nk) <= mu*(A_n) + e/2^n
21:29:27 somiaj: i.e. since mu*(A_n) is finite we can find a cover that is within e/2^n of
                 the acual outer measure.
21:30:22 somiaj: Now we notice that {I_nk}_{n in N, k in N} is a countable cover of 
                 union_n A_n.
21:31:16 somiaj: thus mu*( union_n A_n ) <= sum_n ( sum_k v(I_nk) )
                                         <= sum_n (mu*(A_n) + e/2^n)
                                          = e + sum_n mu*(A_n)
21:31:57 somiaj: since this holds for all e>0, it follows that
                 mu*( union_n A_n ) <= sum_n mu*(A_n) [countabally sub-additive]
1:34:25 somiaj: yea the indicies n and k just run over the natural numbers.
21:36:40 somiaj: Unfortunately in order to generate an acual measure space we can't just use
                 the outer measure over the whole power set. We have to restrict it. So from
                 the outer measure we restrict to a measure. We say that a subset E is
                 measurable (by the outter measure mu*) if
21:37:13 somiaj: mu*(A) = mu*(A intersect E) + mu*(A - E) for all A subset of Omega
21:37:34 somiaj: This is known as the Caratheodory's Characterization of Measurability.
21:38:29 somiaj: In the case of R^n, not all subests in R^n are measurable, so we have to
                 restrict to ones in which make our outer measure countabally-additive [by
                 outer measure we only have sub-additivity]
21:39:33 somiaj: Equivalently we can say E is measurable if and only if
                 mu*(A union B) = mu*(A) + mu*(B) for all A subest E, and
                 B subset Omega - E
21:42:25 somiaj: It can be proven that the collection of subsets which satisify
                 Caratheodory's Characterization form a sigma-field, and the restriction of
                 mu* to this collection is an countabally additive function.
21:42:41 somiaj: Thus we have a measure space.
21:43:40 somiaj: From here is where the detials begin to get into lots of work.
21:44:25 somiaj: but the idea of the lebesgue measure is we find the 'measure' of a set by
                 covering it with n-rectangles and taking the 'smallest' value of all such
21:45:34 somiaj: Though as I mentioned we can do this for any subest of R^n, we cannot
                 create a measure space (since there exist some really bizar subsets of R^n)
21:46:48 somiaj: to prove that this restriction acually forms a sigma-algebra and that mu*
                 under this restriction is countabally additive takes a lot of work and more
                 than we have time for in this siminar. I was just hoping to give the basic
                 idea of how to define the measure
21:47:57 somiaj: Also to prove that there exists non-measurable sets is a theorm from Vitali,
                 which uses zermelo's axiom (variation of the axiom of choice)
21:48:24 kommodore: !
21:48:28 somiaj: kommodore: yes
21:48:51 kommodore: presumbly the sigma-field given is going to be complete wrt to the
21:49:21 somiaj: correct.
21:49:42 somiaj: One intersting result to get a grasp on what the measurable sets acually
                 look like is to start with the borell sets
21:49:54 somiaj: remember the borell sets are the smallest sigma-field containing the open
21:50:13 somiaj: (i.e. all open sets, all closed sets and all countable unions/intersections
                  of such sets)
21:51:05 somiaj: And then to indroduce the concept of a set of measure zero (mu*(A) = 0).
                 Turns out all measure-zero sets and all borell sets are measurable by this
21:52:02 somiaj: further more all measurable sets are within a measure-zero set of a borell
21:52:23 somiaj: though measure zero sets can be quiet bizar, the cantor set is an example
                 of an uncountable set of measure-zero
21:52:59 somiaj: Another useful idea to introduce is what is called G_sigma and F_sigma sets.
21:53:25 somiaj: a G_sigma set is a countable intersection of open sets, and a F_sigma set
                 is a countable union of closed sets.
21:53:52 somiaj: since these sets may neither be open or closed, but must exsist in our
                 borell sigma-field it is nice to give them a name as they become useful in
21:56:51 somiaj: I don't really have anything else prepared and the hour is up, I can try to
                 answer any questions and hope that for people who havent seen measure
                 theory they now have some idea of how the lebesgue measure is created
21:58:31 _llll_: thanks for the seminar, a good intro
21:59:10 drizzd_: !
21:59:13 pyninja: thanks
21:59:53 somiaj: drizzd_: yes
22:00:09 drizzd_: did you mean to say: every measureable set is a union of a borel set and a measure-zero set?

22:01:03 somiaj: yea, measure theory is one of those theories that I think is very
                 intuitive at the start, it is a nice generaliziation of volume in R^n,
                 but the detials of the thoery quickly begin to use a lot of analysis tools
                 so lots of places save it for a graduate course

22:01:51 somiaj: drizzd_: the acual theorem for "A characterization of lebesgue
                 measurability" is
22:02:21 somiaj: 1) E subest R^n is lebesuge measurable if and only if E = G - Z, where G is
                    of type G_sigma and Z is a set of measur zero.
22:02:55 somiaj: or 2) E is lebesgue measurable if and only if E = F union Z where F is of
                       type F_sigma and Z is of measure zero
22:03:20 somiaj: again G_sigma is a countable intersection of open sets, and F_sigma is a
                 countable union of closed sets (both borell sets)
22:03:56 lotekk-: somiaj: thank you
22:04:07 drizzd_: I understand
22:04:44 somiaj: so in that light the lebesgue measurable sets are 'almost' borell sets.
22:05:08 zachk: am i correct in assuming there are more bizarre sets of R^n then measurable
22:06:03 somiaj: zachk: yes, it turns out there are some bizar non-measurable sets out there
22:06:07 Jafet: zachk, somiaj mentioned the Vitali set.
22:06:14 somiaj: is an example of a creation of one
                 for the reals
22:07:00 somiaj: Another example is the sets created in the
22:07:02 somiaj:
22:07:09 drizzd_: and there is the banach-tarski paradox for R^3
22:07:22 somiaj: all the ones I've seen use the axiom of choice to create them, as in they
                 are not things you can easilly visualize
22:08:06 zachk: so as i go higher up in mathematics it will eventually become improbable for
                me to visualize alot of constructs?
22:08:11 kommodore: i think you necessarily need AC to construct them
22:09:47 somiaj: zachk: lots of being able to 'visualize' is a personal issue. Some people
                 visualize certain tyes of constructs better than others, for most there are
                 areas in math that you just have to get use to the proofs of the theorms to
                 get a vague idea of the constructs, for others it seems clear.
22:10:42 FunctorSalad: zachk: this is a particularly bad offender though. most things in
                       higher math aren't *that* non-constructive IMHO
22:11:38 kommodore: well, counterexamples are usually bad offenders...
22:12:00 thermoplyae: _llll_: next weekend i'm at a conference and a wedding
22:12:08 FunctorSalad: kommodore: yeah, and this is a counterexample *and* AC-only ;)
22:13:05 drizzd_: I heard that it has been shown in ZF + a large cardinal axiom that it is
                  impossible to prove the existence of lebesgue-measurable sets in ZF.
22:16:38 toed: apparently ZF + notAC has every subset measurable iirc

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