Prince Rupert's cube: Difference between revisions

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Lifting Theory was first introduced by John von Neumann in his (1931) pioneering paper (answering a question raised by A. Haar),<ref>von Neumann, J.: Algebraische Repräsentanten der Funktionen bis auf eine Menge von Maße Null. J. Crelle '''165''', 109-115 (1931)</ref> followed later by Dorothy Maharam’s (1958) paper,<ref>Maharam, D.: On a theorem of von Neumann. Proc. Amer. Math. Soc. '''9''', 987-995 (1958)</ref> and by A. Ionescu Tulcea and C. Ionescu Tulcea’s (1961) paper.<ref>A. Ionescu Tulcea and C. Ionescu Tulcea: On the lifting property, I., J. Math. Anal. App. '''3''', 537-546 (1961)</ref> Lifting Theory was motivated to a large extent by its striking applications; for its development up to 1969, see the Ionescu Tulceas' work and the monograph,<ref>Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea, ''Topics in the Theory of Lifting'', Ergebnisse der Mathematik, Vol. 48, Springer-Verlag, Berlin, Heidelberg, New York (1969)</ref> now a standard reference in the field. Lifting Theory continued to develop after 1969, yielding significant new results and applications.
As is quite evident from the name, a Combat knife is commonly employed in a war type predicament or for common self-defense purposes. So if you ever do come across oneself in will need of a combat knife, you better have the most effective combat knife possible.<br><br><br><br>An additional factor that we praised about this knife is its really hard sheath. The sheath has received astounding evaluations for being really challenging with constructed in non-slip attributes. In our opinion, this is 1 of the best sheaths and you can certainly have a peace of mind when carrying this knife in the sheath. Probably 1 factor to pay focus to on this knife is that some people obtain the hand guard can cause some discomfort just after [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=Fillet+knives+have+narrow%2C+long%2C+flexible+blades+and+are+utilised+to+remove+the+bones+from+fish.+The+shape+and+flexibility+permit+the+blade+to+simply+move+around+the+bones+and+swiftly+separate+the+flesh.+These+knives+can+also+be+used+to+slice+thin+strips+of+beef+and+chicken+just+before+cooking.+When+deciding+on+a+fillet+knife%2C+you+will+want+to+take+into+account+the+length+of+the+fish+you+will+be+filleting.%0D%0A%0D%0ATruly%2C+this+strategy+will+operate+on+most+fish%2C+so+%5Bhttp%3A%2F%2FWww.Wonderhowto.com%2Fsearch%2Fexperiment%2F+experiment%5D.+Go+in+the+kitchen+and+get+your+electric+knife.+If+it+is+not+a+single+with+a+trigger+on+the+bottom%2C+put+it+back+where+you+located+it.+The+beating+you+will+take+for+making+use+of+the+wrong+type+of+knife+will+not+be+worth+it%2C+trust+me.+Go+to+Wal+Mart+and+get+your+self+a+brand+X+electric+carving+knife.+Just+make+confident+it+has+a+trigger+on+the+bottom%2C+not+a+push+button+operated+by+your+thumb.+The+knife+I+use+cost+me+a+whopping+%249.+No+need+to+have+to+get+the+finest+one.+I+do+not+propose+the+DC+powered+fillet+knives.+We+all+preserve+one+particular+in+the+boat+for+shore+lunches.+But+when+you+are+going+to+clean+a+bunch+of+fish+you+will+appreciate+the+power+of+the+AC+knife.+Besides%2C+the+DC+knives+are+big+and+really+hard+to+handle.%0D%0A%0D%0AI+looked+forever+for+a+fillet+knife+created+from+plain+high+carbon+steel.+I+finally+settled+on+a+Dexter-Russell.+Yes%2C+the+blade+will+rust+if+I+do+not+dry+it%2C+and+apply+a+protectant.+And+it+will+not+hold+an+edge+as+long+as+my+stainless+knives.+However%2C+when+it+comes+time+to+resharpen+it%2C+a+few+strokes+is+all+it+takes+to+place+it+back+to+razor+sharp.+It+really+is+a+pleasure+to+use%2C+and+the+ease+of+sharpening+has+me+sold+on+it.+it%27s+the+only+one%C2%A0I+use+now.+true+testament+to+the+art+of+knife%2C+this+knife+and+tooled+leather+sheath+will+make+a+prized+possession+for+any+recipient.+Trophy+class+fish+are+brood+fish+that+assistance+hold+the+lakes+populated+and+present+chance+for+future+anglers.+Catch%2C+photograph+and+release+is+the+only+way+to+go+with+the+big+catfish.%0D%0A%0D%0AFilleting+is+a+common+technique+of+preparing+fish+for+meals.+With+practice+and+the+correct+knife%2C+filleting+is+seriously+quick.++If+you+loved+this+post+and+you+would+certainly+like+to+obtain+more+information+concerning+%5Bhttp%3A%2F%2FWww.thebestpocketknifereviews.com%2Fbest-fillet-knife-electric-fish-knives-reviews%2F+Electric+fillet+knife+reviews%5D+kindly+visit+the+website.+Right+here+are+a%C2%A0few+video+ideas+to+go+over+these+quick+solutions+and+tactics+about+filleting+a+Walleye%2C+how+to%C2%A0debone+a+Northern+Pike+and+our+favored+a+Northwoods+Fish+Fry%C2%A0from+Badfish+Outdoors.+So+start+off+with+a+firm+cutting+board%2C+a+sharp+knife+with+a+comfy+grip+and+a+blade+of+just+the+correct+length.+Add+a+firm%2C+fresh+fish%2C+wiped+dry.+Always+reduce+against+bone%2C+or+if+you+prefer+skinless+fillets%2C+the+skin%2C+and+two+tidy+fillets+pretty+much+magically+seem.+Waste+is+minimal.+Attempt+to+fillet+fish+with+a+dull+heavy+blade+and+your+outcome+may+well+appear+more+like+lace+curtains.+Victorinox+47513+6-Inch+Flex+Boning+Knife+with+Fibrox+Deal+with+Smaller+Knives&create=Create extended] use in the woods for debrake or carve woods. But it all depends on your grip gesture and the duration of a single use. three. Buck 119BR Specific Fixed Blade Knife.<br><br>To push this point, I will inform a correct story of when I went to Manilla for the initial time in 1981. I was watching a group of young guys training in their Filipino fighting arts of stick and knife. They were dressed in normal clothing and going for it. An older gentleman was sitting on the similar brick fence that I was sitting on. At one point, he got up and approached these young guys who started to attack him! Just about to jump up and assistance when this old man kicked the shyte out of the young guys! And remain pals with some of these students.<br><br>Filet knives have comparable requirements, but for extremely unique causes. A filet knife wants to be versatile and does not tolerate brittleness quite properly, so after once more toughness is a essential attribute. Carbon content material when once again tends to be in the intermediate variety (.6% to .8%), since a lot more carbon may possibly lead to unacceptable brittleness. If you cherished this article and you would like to be given more info concerning [http://www.thebestpocketknifereviews.com/best-fillet-knife-electric-fish-knives-reviews/ berkley electric Fillet knife reviews] kindly visit our own web-site. Filet knives tend to be heat treated tends in the mid 50s Rc to give a excellent compromise involving toughness and the potential to take a fine edge (the narrow blade profile on most filet knives makes them quick to re-sharpen promptly, so edge retention is not typically a concern).<br><br>The Pathfinder Combat Tactical Knife from United Cutlery may well be a much better choice as a camp tool or an emergency knife for the car or truck trunk. It is huge and impressive, but it really is heavy. The compass comes up to military requirements, but only if you happen to be into WWII surplus. With the curved blade design, this knife is wonderful for cutting, slicing, skinning, or other delicate works. It is also worth mentioning the light weight but sturdy anodized aluminum deal with. And do not worry about slipping for the reason that it is created of metal. The thin rubber inserts on the manage deliver very good grip to the knife. Overall, this is one of the very best tactical folding knife in its class. We hugely advise it! five. Cold Steel 17T Kobun Tanto Fixed this is by far a very nicely performed copy. $ 50.00<br><br>Folding knives are seldom if ever made mainly for use as fighting knives or combat knives. Nevertheless, many armies and military organizations have issued folding "utility" knives that had been not intended to be applied as weapons, but which had tactical capabilities that appealed to military personnel as well as civilians. This involves the German Mercator "Black Cat" folding utility knife,  35  the German Luftwaffe Fallschirmjäger-Messer , the British Ibberson WWII gravity knife, and the U.S. WWII M2 automatic pressbutton utility knife, also intended for use by parachute troops and flight crews.<br><br>The SAS nicely demonstrates the "highlights" inherent in the Bowie Style. Visible on initially glance are the Bowie "look," demonstrating the one of a kind proportion among handle and blade, the ever present hand guard, and the classic Bowie blade, which can run up to 12 inches in length in a contemporary knife. In the course of this report, we have presented the major [http://tahoe.sourceforge.net/wiki/index.php/What_Is_The_Best_Electric_Fish_Fillet_Knife influences] in current knife style, and have demonstrated these influences with numerous examples. You are nicely on the road toward getting in a position to "study" your opponent. The rest is up to you! I have observed a severe more than emphasis on defense just before closing and a serious lack of emphasis immediately after closing - either one will get you mauled, if not killed. That is the difference in between fighting and combat.<br><br>I got a single of these as a present to myself and I have to say wow. It is a wonderful knife with an incredible really feel. The sheath is innovative and trustworthy. It is a little challenging to put back in but the the draw is effortless (right after all, that's the action that matters most) Considering that it was discontinued, I am afraid to use it because if I lost it I would be unable to get a new one particular. Glad I got it when I did!! I bought this knife for a second tour in afghan. Feels sturdy and th sheath is incredibly strong. I messed the sheath up [http://sss.chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=In-Fisherman+has+long+promoted+the+practice+of+Selective+Harvest+%2C+which+encompasses+the+practice+of+maintaining+some+fish+selectively+as+a+matter+of+conscience+and+conservation%2C+and+for+the+reason+that+fish+are+nutritious%2C+scrumptious%2C+and%E2%80%94when+harvested+wisely%E2%80%94they%27re+an+infinitely+renewable+resource.%0D%0A%0D%0ABlues+and+flatheads+as+a+common+rule+have+a+a+lot+cleaner+diet+regime+feeding+primarily+as+predator+fish+%28they+eat+a+lot+of+live+stuff%29+and+do+not+spend+as+much+time+scavenging.+That+is+aspect+of+it+but+there%27s+also+the+fact+that+they+are+completely+various+species+of+fish+Catfish+Amongst+Five+and+Ten+Pounds+The+fish+will+keep+for+a+Really+long+time+frozen+like+this+and+will+not+get+freezer+burn.+I%27ve+kept+fillets+for+properly+over+a+year+without+difficulties.+What+Do+You+Do+With+Fish+Carcasses%3F+This+function+a+Stainless+Steel+Finger+Guard+for+your+security.+This+will+support+to+avoid+the+injuries+triggered+by+slippage.+As+often%2C+Buck+Silver+Creek+ideal+filet+knife+is+backed+by+the+Buck%27s+Forever+Warranty+It+really+is+a+terrific+tool+to+add+with+your+fishing+kit.+Zwilling+J.A.+Henckels+Knives%0D%0A%0D%0AKopp+earned+%E2%80%9CBest+Fillet+Knife%E2%80%9D+honors+at+the+2010+Skilled+Knifemakers+Association+Show+with+his+damascus+fillet+knife.++If+you+loved+this+informative+article+and+you+would+like+to+receive+details+about+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fbest-fillet-knife-electric-fish-knives-reviews%2F+fish+fillet+Knife+Kit%5D+please+visit+our+web-page.+He+strayed+from+traditional+stainless+steel+for+his+award-winning+knife%2C+opting+instead+for+a+damascus+of+1084+and+15N20+carbon+steels.+The+knife+was+a+Rapala+7%E2%80%B3+Fish%27n+Fillet+%28link+is+to+Amazon%29.+Dave+uses+it+for+fundamental+fish+filleting%2C+and+I+both+to+eliminate+the+skin+from+the+fillets+and+also+for+boning+other+meat+%28you+can+see+me+applying+it+in+my+video+on+how+to+bone+a+chicken+breast+%29.+It+is+infinitely+far+better+than+the+%E2%80%9Cfillet+and+boning+knife%E2%80%9D+that+came+with+my+other+knives+%E2%80%94+and+improved+however%2C+it%27s+also+less+costly%21+Understand+how+to+do+the+similar+point+with+a+fantastic+Fillet+Knife+like+the+14%22+Kershaw+that+Green+tops+was+promoting+for+%2414%21%0D%0A%0D%0AAmerican+Angler+Pro+Knife+is+a+should+really-study+electric+fillet+knife+reviews.+As+we+talked+about+in+the+beginning+it%27s+essential+to+purchase+a+electric+ideal+fillet+knives+if+you+catch+a+lot+of+fishes.+The+very+best+model+for+this+task+is+the+one+particular+described+right+here.+Wusthof+Gourmet+7-Inch+Fish+Fillet+Knife+with+Leather+Sheath+All+round+this+a+tremendous+worth.+You+could+acquire+five+or+six%2C+maybe+as+numerous+as+ten+Silver+Creek+Fillet+Knives+for+the+expense+of+the+greatest+fish+fillet+knife+dollars+can+purchase.+When+it+comes+down+to+it%2C+you+must+get+the+Silver+Creek+and+devote+your+cash+on+extra+vital+pieces+of+gear.+Greatest+Electric+Fish+Fillet+Knife+Buck+Silver+Creek++Folding+Fillet+Knife+Curved%2C+semi+stiff+fish+fillet+knife+Bubba+Blade+7-Inch+Tapered+Flex+Fillet+Knife+Check+All+Renowned+Fillet+Knives+on%0D%0A%0D%0A1.Insert+the+tip+of+the+fillet+knife+into+the+gill+plate.+This+is+just+behind+the+pectoral+fin+on+the+side+of+the+fish.+Have+the+blade+of+the+knife+facing+towards+the+major+of+the+fish+for+the+next+step.+For+cleaning+both+significant+and+small+fish%2C+get+a+fantastic+filet+knife.+The+most+economical+use+of+the+fish+frequently+turns+out+to+be+a+neatly+cut+filet.+A+knife+that%27s+too+stiff%2C+also+small%2C+or+as+well+brief+wastes+a+lot+of+fish.+For+the+larger+catch+you+are+going+to+need+a+full+sized+flexible+blade+like+our+decision+for+the+greatest+fillet+knife+%E2%80%94+Queen+Cutlery%27s+Fishing+and+Filet+Knife+See+All+of+Our+Fishing+Knife+Critiques+Below+That+which+tends+to+make+a+very+good+knife+makes+a+fantastic+fillet+knife.%E2%80%9D+Fish+Fillet+knifes%2C+5%22+narrow+flexible+Premium+good+quality+set+of+fish+fillet+knives+Dexter+Russell+Terrific+knife%0D%0A%0D%0A%5Bhttp%3A%2F%2Fbrowse.deviantart.com%2F%3Fqh%3D%26section%3D%26global%3D1%26q%3DElectric%2Bfillet+Electric+fillet%5D+knives+can+overheat+when+cleaning+a+lot+of+fish.+We+applied+a+Raytek+non-get+in+touch+with+thermometer+to+measure+temperatures+of+the+motor+housings+as+fish+were+becoming+cleaned.+For+instance%2C+the+Mister+Twister+Electric+Fisherman+reached+a+surface+temperature+of+108+degrees+soon+after+filleting+about+18+fish.+Most+kitchen+knives+have+a+20-degree+edge+bevel.+For+filleting%2C+an+edge+bevel+of+12+to+15+degrees+per+side+offers+a+clean%2C+quick+reduce+and+reasonable+resistance+to+dulling.+If+you+don%27t+reduce+bone%2C+monofilament+or+bait+with+your+fillet+knife%2C+a+15-degree+bevel+will+serve+you+fine.+The+oyster+knife+that+shucks+like+no+other+and+turns+heads+nationwide%2C+named+Overall+Winner+of+the+Created+in+the+South+Awards+2011+by+Garden+%26+Gun.+Fillet+Storing+%26+Freezing%C2%A0Tips&create=Create attempting] to clean dirt out of it and lost the screws. Gerber sent me a complimentary sheath!! I have been a customer since I joined the Army in 2008 and this gesture will make me continue to by your solutions! A double edged blade would be preferable.<br><br>We do not want to see soldiers going to war or on assignments with knives that are labeled “tactical” for the reason that they are painted or produced with materials that are black. Any soldier, or person for that matter, really should have a Combat Knife no matter whether it has the word “tactical” in the title or not that will offer you with the overall performance and adaptability for any kind of combat/survival situation. Complete Tang and Partial Tang Fixed blade knives have the blade fixed in its spot and do not fold or move in any way. Because this type of knife does not move it is the only type of knife that has a tang where folding or sliding blade knives do not. A screwdriver or ice pick can conveniently be shoved through a car or truck door, but you would not carry either one into combat. Tac Knife
 
A '''lifting''' on a [[measure space]] (''X'', Σ, μ) is a linear and multiplicative inverse
 
:<math> T:L^\infty(X,\Sigma,\mu)\to \mathcal L^\infty(X,\Sigma,\mu)</math>
 
of the quotient map
 
: <math>\begin{cases}\mathcal L^\infty(X,\Sigma,\mu)\to L^\infty(X,\Sigma,\mu) \\
f\mapsto [f]\end{cases}</math>
 
In other words, a lifting picks from every equivalence class [''f''] of bounded measurable functions modulo negligible functions a representative&mdash; which is henceforth written ''T''([''f'']) or ''T''[''f''] or simply ''Tf'' &mdash; in such a way that
 
:<math>T(r[f]+s[g])(p)=rT[f](p) + sT[g](p), \qquad \forall p\in X, r,s\in \mathbf R;</math>
:<math>T([f]\times[g])(p)=T[f](p)\times T[g](p), \qquad \forall p\in X;</math>
:<math>T[1]=1.</math>
 
Liftings are used to produce [[Disintegration theorem|disintegrations of measures]], for instance [[conditional probability distribution]]s given continuous random variables, and fibrations of Lebesgue measure on the level sets of a function.  
<!--
==Properties of liftings==
A lifting is necessarily positive:
:<math>[f]\ge0\implies T[f]\ge0 \mathrm{\ (since\ } [f] \mathrm{\ is\ a\ square)} </math>
 
and an isometry:<ref>The ''essential supremum'' of a class [''f''] of μ-measurable functions is the smallest number α for which the set [''f'' > α] is μ-negligible.</ref>
 
:<math> \big\Vert T[f]\big\Vert_\infty:= \sup_{p\in X}\,\big|T[f](p)\big|=\mathrm{ess.sup}\,\big|[f]\big|.</math>
 
For every point ''p'' in ''X'', the map <math>[f]\mapsto T_pf:= T[f](p)</math> is a character<ref name=character> A ''character'' on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.</ref> of ''L''<sup>∞</sup>(''X'', Σ, μ).
-->
 
==Existence of liftings==
<blockquote>'''Theorem.''' Suppose (''X'', Σ, μ) is complete.<ref>A subset ''N'' ⊂ ''X'' is locally negligible if it intersects every integrable set in Σ in a subset of a negligible set of Σ. (''X'', Σ, μ) is ''complete'' if every locally negligible set is negligible and belongs to Σ.</ref> Then (''X'', Σ, μ) admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in Σ whose union is ''X''.
 
In particular, if (''X'', Σ, μ) is the completion of a σ-finite<ref>i.e., there exists a countable collection of integrable sets &ndash;sets of finite measure in Σ&ndash; that covers the underlying set ''X''.</ref> measure or of an inner regular Borel measure on a locally compact space, then (''X'', Σ, μ) admits a lifting.</blockquote>
 
The proof consists in extending a lifting to ever larger sub-σ-algebras, applying [[Doob's martingale convergence theorems|Doob's martingale convergence theorem]] if one encounters a countable chain in the process.
 
<!--  
Here are the details. Henceforth write ''Tf'' := ''T''[''f''] = ''T''([''f'']). (Σ, μ) is σ-finite if there exists a countable collection of sets of finite measure in Σ whose union has negligible complement. This permits a reduction to the case that the measure μ is finite, in fact, it may be taken to be a probability. The proof uses [[Zorn's lemma]] together with the following order on pairs <math>(\mathfrak A,T_{\mathfrak A})</math> of sub-σ-algebras <math>\mathfrak A</math> of Σ and liftings <math>T_{\mathfrak A}</math> for them: <math> (\mathfrak A,T_{\mathfrak A})\le(\mathfrak B,T_{\mathfrak B}) </math> if <math>\mathfrak A\subseteq\mathfrak B</math> and <math>T_{\mathfrak A}</math> is the restriction of <math>T_{\mathfrak B}</math> to <math>L^\infty(X,\mathfrak A,\mu)</math>. It is to be shown that a chain <math>\mathfrak C</math> of such pairs has an upper bound, and that a maximal pair, which then exists by Zorn's lemma, has Σ for its first entry.
 
If <math>\mathfrak C</math> has no countable [[Cofinal (mathematics)|cofinal]] subset, then the union <math>\mathfrak U:=\bigcup\{\mathfrak A:\,(\mathfrak A,T_{\mathfrak A})\in\mathfrak C\} </math> is a σ-algebra and there is an obvious lifting <math>T_{\mathfrak U}</math> for it that restricts to the liftings of the chain; <math>(\mathfrak U,T_{\mathfrak U})</math> is the sought upper bound of the chain.
 
The argument is more complicated when the chain <math> \mathfrak C</math> has a countable cofinal subset <math>\left\{(\mathfrak A_n,T_{\mathfrak A_n}),n=1,2,\ldots\right\}</math>. In this case let <math>\mathfrak U</math> be the  [[Sigma-algebra|σ-algebra generated]] by the union <math>\bigcup\{\mathfrak A_n:\,n=1,2,\ldots\} </math>, which is generally only an [[Field of sets|algebra of sets]]. For the construction of <math>T_{\mathfrak U}</math> it is convenient to identify a set ''A'' ⊆ ''X'' with its indicator function and to write <math>TA:=TI_A=T[I_A]</math>. For <math>A\in\mathfrak U</math> let ''A<sub>n</sub>'' denote the [[conditional expectation]] of ''A'' under <math>\mathfrak A_n</math>. By [[Doob's martingale convergence theorems|Doob's martingale convergence theorem]] the set θ(''A'')of points where ''A<sub>n</sub>'' converges to 1 differs negligibly from ''A''.
 
Here are a few facts that are straightforward to check (some use the completeness and finiteness of <math>(X,\mathfrak U,\mu)</math>):
:<math> \tau:=\{\theta(A)\setminus N \ : \ A\in\mathfrak U, \mu(N)=0\}\subset\mathfrak U</math>
is a topology whose only negligible open set is the empty set and such that every <math> A=I_A\in\mathfrak U</math> is almost everywhere continuous, to wit, on <math> A\cap\theta(A)</math> and on <math> A^c\cap\theta(A^c)</math>. Then every <math>f \in\mathcal L^\infty(X,\mathfrak U,\mu)</math>, being the uniform limit of a sequence of step functions over <math>\mathfrak U</math>, is almost everywhere continuous in this topology. For ''p'' in ''X''
 
:<math> I_p:=\{[f]: f\mathrm{\ is\ continuous\ at\ }p\mathrm{\ and\ }f(p)=0\}.</math>
 
is a proper ideal of <math> L^\infty(X,\mathfrak U,\mu)</math>, contained (by another application of Zorn's lemma) in some maximal proper ideal <math> J_p\subset L^\infty(X,\mathfrak U,\mu)</math>, which has codimension 1. The quotient map <math>L^\infty(X,\mathfrak U,\mu)\to L^\infty(X,\mathfrak U,\mu)/J_p</math> can be viewed as a character<ref name=character/>''T<sub>p</sub>''. Defining
 
:<math> \left(T_{\mathfrak U}[f]\right)(p):=T_p[f]\;\;,\;\;\;\;\;\;p\in E,</math>
 
provides the upper bound <math>(\mathfrak U,T_{\mathfrak U})</math> for the chain <math>\mathfrak C</math>.
 
In either case the chain <math> \mathfrak C</math> therefore has an upper bound. By Zorn's lemma there is a maximal pair <math>(\mathfrak U,T_{\mathfrak U})</math>,
and a small additional calculation shows that <math> \mathfrak U=\mathfrak F</math>. END OF DETAILED PROOF-->
 
== Strong liftings ==
Suppose (''X'', Σ, μ) is complete and ''X'' is equipped with a completely regular Hausdorff topology τ ⊂ Σ such that the union of any collection of negligible open sets is again negligible &ndash; this is the case if (''X'', Σ, μ) is σ-finite or comes from a Radon measure. Then the ''support'' of μ, Supp(μ), can be defined as the complement of the largest negligible open subset, and the collection ''C<sub>b</sub>''(''X'', τ) of bounded continuous functions belongs to <math> \mathcal L^\infty(X,\Sigma,\mu)</math>.
 
A '''strong lifting''' for (''X'', Σ, μ) is a lifting
:<math> T:L^\infty(X,\Sigma,\mu)\to \mathcal L^\infty(X,\Sigma,\mu)</math>
such that ''T''φ = φ on Supp(μ) for all φ in ''C<sub>b</sub>''(''X'', τ). This is the same as requiring that<ref>''U'', Supp(μ) are identified with their indicator functions.</ref> ''TU'' ≥ (''U'' ∩ Supp(μ)) for all open sets ''U'' in τ.
 
<blockquote>'''Theorem.''' If (Σ, μ) is σ-finite and complete and τ has a countable basis then (''X'', Σ, μ) admits a strong lifting.</blockquote>
 
'''Proof.''' Let ''T''<sub>0</sub> be a lifting for (''X'', Σ, μ) and {''U''<sub>1</sub>, ''U''<sub>2</sub>, ...} a countable basis for τ. For any point ''p'' in the negligible set
 
:<math>N:=\bigcup\nolimits _n \left\{p\in \mathrm{Supp}(\mu): (T_0U_n)(p)<U_n(p) \right\}</math>
 
let ''T<sub>p</sub>'' be any character<ref name=character>A ''character'' on a unital algebra is a multiplicative linear functional with values in the coefficient field that maps the unit to 1.</ref> on ''L''<sup>∞</sup>(''X'', Σ, μ) that extends the character φ ↦ φ(''p'') of ''C<sub>b</sub>''(''X'', τ). Then for ''p'' in ''X'' and [''f''] in ''L''<sup>∞</sup>(''X'', Σ, μ) define:
 
:<math> (T[f])(p):= \begin{cases} (T_0[f])(p)& p\notin N\\
T_p[f]& p\in N.
\end{cases}</math>
 
''T'' is the desired strong lifting.
 
==Application: disintegration of a measure==
Suppose (''X'', Σ, μ), (''Y'', Φ, ν) are σ-finite measure spaces (μ, ν positive) and π : ''X'' → ''Y'' is a measurable map. A '''disintegration of μ along π with respect to ν''' is a slew <math>Y\ni y\mapsto \lambda_y</math> of positive σ-additive measures on (''X'', Σ) such that
 
#λ<sub>''y''</sub> is carried by the fiber <math>\pi^{-1}(\{y\})</math> of π over ''y'':
:::<math> \{y\}\in\Phi\;\;\mathrm{ and }\;\; \lambda_y\left((X\setminus \pi^{-1}(\{y\})\right)=0 \qquad \forall y\in Y</math>
#for every μ-integrable function ''f'',
:::<math> \int_X f(p)\;\mu(dp)= \int_Y \left(\int_{\pi^{-1}(\{y\})}f(p)\,\lambda_y(dp)\right) \nu(dy) \qquad (*)</math>
::in the sense that, for ν-almost all ''y'' in ''Y'', ''f'' is λ<sub>''y''</sub>-integrable, the function
:::<math> y\mapsto \int_{\pi^{-1}(\{y\})} f(p)\,\lambda_y(dp) </math>
::is ν-integrable, and the displayed equality (*) holds.
 
[[Disintegration theorem|Disintegrations]] exist in various circumstances, the proofs varying but almost all using strong liftings. Here is a rather general result. Its short proof gives the general flavor.
 
<blockquote>'''Theorem.''' Suppose ''X'' is a Polish<ref>A separable space is ''Polish'' if its topology comes from a complete metric. In the present situation it would be sufficient to require that ''X'' is ''Suslin'', i.e., is the continuous Hausdorff image of a polish space.</ref> space and ''Y'' a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a σ-finite Borel measure on ''X'' and π : ''X'' → ''Y'' a Σ, Φ&ndash;measurable map. Then there exists a σ-finite Borel measure ν on ''Y'' and a disintegration (*).
 
If μ is finite, ν can be taken to be the pushforward<ref>The ''pushforward'' π<sub>∗</sub>μ of μ under π, also called the image of μ under π and denoted π(μ), is the measure ν on Φ defined by <math>\nu(A):=\mu\left(\pi^{-1}(A)\right)</math> for ''A'' in Φ.</ref> π<sub>∗</sub>μ, and then the λ<sub>''y''</sub> are probabilities.</blockquote>
 
'''Proof.''' Because of the polish nature of ''X'' there is a sequence of compact subsets of ''X'' that are mutually disjoint, whose union has negligible complement, and on which π is continuous. This observation reduces the problem to the case that both ''X'' and ''Y'' are compact and π is continuous, and ν = π<sub>∗</sub>μ. Complete Φ under ν and fix a strong lifting ''T'' for (''Y'', Φ, ν). Given a bounded μ-measurable function  ''f'', let <small><math>\lfloor f\rfloor</math></small> denote its conditional expectation under π, i.e., the [[Radon–Nikodym theorem|Radon-Nikodym derivative]] of<ref>''f''μ is the measure that has density ''f'' with respect to μ</ref> π<sub>∗</sub>(''f''μ) with respect to π<sub>∗</sub>μ. Then set, for every ''y'' in ''Y'', <math>\lambda_y(f):=T(\lfloor f\rfloor)(y).</math> To show that this defines a disintegration is a matter of bookkeeping and a suitable Fubini theorem. To see how the strongness of the lifting enters, note that
 
:<math> \lambda_y(f\cdot\varphi\circ\pi)=\varphi(y) \lambda_y(f) \qquad \forall y\in Y, \varphi\in C_b(Y), f\in L^\infty(X,\Sigma,\mu)</math>
 
and take the infimum over all positive φ in ''C<sub>b</sub>''(''Y'') with φ(''y'') = 1; it becomes apparent that the support of λ<sub>''y''</sub> lies in the fiber over ''y''.
 
== References ==
 
<references />
 
[[Category:Measure theory]]

Latest revision as of 08:44, 15 June 2014

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