Talk:Probability measure

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Probability measures are not real valued functions defined on a domain

because in general they are not functions.71.206.193.135 (talk) 21:17, 31 March 2012 (UTC)

For example, the Dirac delta mass measure is a probability measure, but it is not a function. 71.206.193.135 (talk) 21:20, 31 March 2012 (UTC)

The Dirac delta measure is indeed a function: it is a function from a sigma algebra to {0,1}. — Preceding unsigned comment added by 67.193.102.154 (talk) 00:30, 27 December 2013 (UTC)

In some cases that is true, although Carnap's book does say function, as does this one. So we can probably say an assignment of a probability, etc. History2007 (talk) 21:32, 31 March 2012 (UTC)

Definition of probability missing

Hey, haven't you authors noticed that you haven't given any examples of actual probability measures?

BCG999 Out. (talk) 16:36, 19 November 2012 (UTC)

nominate for deletion

this page should have never been created. the entire point of the lebesgue measure is to provide a mathematically legitimate "probability measure" (which is not legitimate).

further, the user (user:VanishedUserABC) who created this page did so after an expert in this area passed away (https://web.archive.org/web/20100324055343/http://www.nationalpost.com/news/canada/story.html?id=2439178&p=2).

this aforementioned expert produced a methodology that was supposed to reveal the properties of the lebesgue integral (see http://cs.nyu.edu/~roweis/lle/).

this expert would have rejected the creation of this page, on the basis that the lebesgue measure is sufficient.

this entire situation is so alarming that i hope someone can help. User:Materialscientist?? anyone?? user:VanishedUserABC (History2007) disappeared around the time word was spreading in the area that roweis' LLE had been a success (see http://imgur.com/bhAQPCK). i am very disconcerted about this user's extremely convenient timing in creating this page, and also disappearing... 174.3.155.181 (talk) 06:40, 3 April 2016 (UTC)

Perhaps an argument could be made to merge probability measures with measure (mathematics) as a probability measure is merely a measure ${\displaystyle \mu }$ with total measure ${\displaystyle \mu (X)=1}$ (on a measurable space ${\displaystyle (X,{\mathcal {F}})}$). The Lebesgue measure is a very useful measure, but it is not the only measure, and except if restricted to the unit interval is not a probability measure, i.e., ${\displaystyle \lambda (\mathbb {R} )=\infty \neq 1}$. In the end, even if all one ever needs is the Lebesgue measure, the usefulness of a specific measure in practice does not invalidate a definition. Zfeinst (talk) 13:10, 3 April 2016 (UTC)
do not try and extend an olive branch like this again. this page does not need to exist. the lebesgue measure was intended to be a general measure that captured the underlying regime (hidden variables) responsible for generating the observed output. i am not saying the attempts at alternative measures weren't admissible prior to demonstrating the lebesgue measure's existence in nature, but now that the latter has been shown, the former cannot possibly be admissible. this is because our perception/vision is continuous; thus, if we use mathematics (lebesgue measure) to account for the infinite-dimensional nature of our observations, which then produces fMRI data revealing electrical activity responsible for generating the original input (observed blood-oxygen-level dependent contrast), we have shown that this measure supercedes all others (as lebesgue intended).
If we assume for the moment (even though I disagree with your unreferenced statement) that the Lebesgue measure is the only measure that matters, then: 1. Why single out the page on probability measures and not measure (mathematics) more generally? and 2. How do you define the Lebesgue measure on a general measurable space ${\displaystyle (X,{\mathcal {F}})}$ where ${\displaystyle {\mathcal {F}}}$ is a sigma algebra over the space ${\displaystyle X}$? The Lebesgue measure is defined in Euclidean space with the Lebesgue sigma algebra (though often the Borel sigma algebra is chosen instead), but many more spaces exist. The simplest to consider is a finite probability space. Consider for instance flipping a single coin: ${\displaystyle X=\{H,T\}}$ and ${\displaystyle {\mathcal {F}}=\{\emptyset ,\{H\},\{T\},X\}}$. By the axioms of probability measures I can define any probability measure ${\displaystyle P:{\mathcal {F}}\to [0,1]}$ so that ${\displaystyle P(\{H\})+P(\{T\})=1}$ and ${\displaystyle P(\emptyset )=0}$. A fair coin is when ${\displaystyle P(\{H\})=P(\{T\})=1/2}$, but I can always have a weighted coin instead to favor heads over tails (thus associating a different probability measure). If the Lebesgue measure supersedes all others, how do you define it in this context on this choice of measurable space? If we wanted a more challenging example, just consider the theorem that there is no infinite-dimensional Lebesgue measure on an infinite-dimensional Banach space.
boy i'm sure you felt smart writing that, didn't you? what part of "probability measure is not a measure (mathematics)" are you having trouble understanding? latex on a discrete example, which isn't the continuous example i asked for, is only furthering my suspicion of your shakey foundations.
2. how do i define it? draw a line on a piece of paper. now, tell me, how many points exist along that line? the cantor set clearly demonstrates that there are an infinitely uncountable number of points along this line (even if it does not extend beyond the page on which it is drawn).
we know that the Lebesgue measure assigns value 1 (b-a) to the [0,1] interval, even if it has an infinitely uncountable numbers. now, notice that any number (singleton) on this interval will have measure zero.
indeed, even the set of *rational numbers* has measure ZERO (did you get that?) on the [0,1] interval. your field "mathematical finance" implicitly uses only the rationals by relying on calculation devices to produce BOTH the measurements and analysis.
contrast this with MRI, which does NOT discretise the original measurements (they may be discretised as a result of digitising the data, but the measurements themselves are continuous). the MRI data therefore can invite an interpretation that uses the Lebesgue measure.
lastly, stop using toy examples of heads or tails, which is a discrete state space (two outcomes), when your area is claiming to produce real-valued (infinite dimensional) security/bond prices using alleged measure (mathematics).
you're asking for the application of the Lebesgue measure to a space which has two possible outcomes, and therefore two values on the [0,1] interval. H would occupy exactly half of the interval (T the other half) because we know P({H})=0.5, and thus on the unit interval it must have an *area* of exactly half (assuming the height is one, the rectangle for {H} and {T} is [0,0.5)*[0,1] and [0.5,1]*[0,1]).
i don't see where the need for this there is no infinite dimensional lebesgue measure comes in. the interval itself is infinite dimensional by any of Cantor's arguments, and the lesbesgue measure (after calculating it, which is very hard and non trivial btw) is a real-valued number that "contains" all of the information necessary to integrate the contour along the [0,1] interval. so, the lebesgue measure could be something like 3.23123905812034581234588458354 or something, but if you unpack it to binary, and apply some constraints, it can be re-integrated to procure the original contour (using eigenvalue algorithms that allow you to extract the orthogonal basis, which act as the axes, which we can then use to integrate along another dimension).
many mathematicians, and i mean MANY, have waited decades for a contribution like the one i mentioned. i will therefore refrain from stating much more about the geometry because many of the aforementioned individuals spent years doing "boring" math that had no "real world application" until now (analysing the MRI data).
what i will say is that there is an inherent geometry in the binary form of numbers. if you think about it, any irrational/rational value from the 0,1 interval has a *unique* binary representation, which has geometry.
Hermann Minkowski was stating as much when he founded Geometry of Numbers, but many of these thinkers were too far ahead of their time. we have a lot of mathematical discovery to do, which is exciting because we want to provide you the answers regarding the geometry of numbers and how they pertain to the lebesgue measure.
My "olive branch" still holds, since a probability measure is just a measure with the measure of the whole space being 1. That does not necessarily deserve its own page. However, my statement is for the general Wikipedia audience, not just to user:174.3.155.181, as we should have a consensus before making any such move. Zfeinst (talk) 19:41, 3 April 2016 (UTC)
no, a probability measure is not "just a measure with the measure of of the whole space being 1" (THINK ABOUT WHAT YOU"RE SAYINg, LADDY! this is why you shouldn't be touching such sophisticated concepts!!). the entire point of the lebesgue measure being 1 is to act as an Aleph number describing the size. the size, however, doesn't accurately capture the infinite uncountable nature of the [0,1] interval. further. the reason the measure is "1" is because _ANY_ pattern that exists, no matter how complex it looks, how far it extends (can go from 0 to infinity for all i care) can be "scaled down" to fit on the [0,1] interval because it has an infinitely uncountable number of values.
i seriously grow concerned about your knowledge of mathematics. the fact that you keep thinking this "probability measure is just a measure with a measure of the whole space being 1" is so wrong it is unbelievable. i explained where the number 1 came from (b-a, subtracting left endpoint of the interval, 0, from the right part, 1). even though it is one, it still represents the "size" of an interval that contains an infinitely uncountable (cantor set) number of values (the real line). i hope this helps you understand why you are wrong... 174.3.155.181 (talk) 22:31, 3 April 2016 (UTC)
Here are some references for you for probability measures (in increasing order of authoritative level):
* [1] Wolfram spells it out very explicitly for you;
* [2] Don't like Wolfram, how about the Encyclopedia of Math? Though you'll have to check the definition of a measure yourself, but again it is there;
* [3] And why not go to the source: "Foundations of the Theory of Probability" by Kolmogorov. Zfeinst (talk) 22:48, 3 April 2016 (UTC)
none of the links you've provided defend your claims. you are simply making weak arguments and sheepishly appeal to authority. you do realise that one of my mentors was a grandstudent of Yakov Sinai, right? the space in which financial market data is collected is NOT measurable. you guys do not "measure" anything. these values are not determined by natural causes at all. they are determined by humans. you cannot equate motion in the real world (a ball falling from height h) to movements in price (asset/stock bond price falls h), as one follows the laws of nature and the other does not.
further, why do you keep deflecting to kolmogorov? you do realise his approach to probability was axiomatic and relies on the lebesgue measure, right? so you're inherently weakening your arguments by deferring to the great kolmogorov. you should be careful. kolmogorov's axioms go hand in hand with the lebesgue measure. you clearly do not understand the set theory concepts necessary to construct the probability measure he is referring to (which is the lebesgue measure). the lebesgue measure *is* the probability measure, it's just hard to demonstrate.. REALLY hard. that's why my colleague killed himself (it took 15 years for someone to show the power of his method. he was funded by the canadian institute for advanced research and the tandem paper that explains what Locally Linear Embedding(http://science.sciencemag.org/content/290/5500/2323) was attempting to do is here: http://science.sciencemag.org/content/290/5500/2268 essentially human beings' perception is continuous. which means that when we inherently perceive infinitely-uncountable things (like drawing a line segment on paper) as discrete. it's the same reason why i said mathematical finance must be called quantitative finance, because we are inherently identify things discretely (even if they are not).

hope this helps. i think we're just going to go into a circular argument after this. i feel i did a great job explaining why the lebesgue measure has a value of 1, and why this deceptively simple number grossly oversimplifies the complicated nature of the interval that it is (abstractly) quantifying. i wish terrance tao had wiki. cheers.

I will give you one more reference then for the definition for probability measures since you (rightfully) respect Yakov Sinai. In his book "Theory of Probability and Random Processes" (coauthored with Leonid Koralov), Definition 1.15: "A measure ${\displaystyle P}$ on a measurable space ${\displaystyle (\Omega ,{\mathcal {F}})}$ is called a probability measure or a probability distribution if ${\displaystyle P(\Omega )=1}$."
This is ignoring the argument that you are apparently having, which is seemingly solely against math finance; probability theory and stochastic processes are a field unto themselves and are not solely within the realm of financial math. As you said, there is not point in continuing this any further since you are not considering the literature of the pure mathematics: Yes the Lebesgue measure is an important measure on Euclidean space, but it is not the only measure (here is another measure: Let ${\displaystyle X:\mathbb {R} \to \mathbb {R} }$ be a Gaussian random variable and let ${\displaystyle \lambda :{\mathcal {B}}(\mathbb {R} )\to \mathbb {R} _{+}}$ be the Lebesgue measure then ${\displaystyle \mu (A):=\lambda (\{\omega \in \mathbb {R} \;|\;X(\omega )\in A\})\;\forall A\in {\mathcal {B}}(\mathbb {R} )}$ is a measure that is not equal to the Lebesgue measure by definition). Additionally, just to recall an earlier point, the Lebesgue measure cannot be generalized to an infinite dimensional Banach space (cf. infinite-dimensional Lebesgue measure), and no, the interval is not infinite dimensional - it has uncountably infinite points but a point and a dimension are not equivalent. Zfeinst (talk) 02:31, 4 April 2016 (UTC)

you have clearly shown yourself inept. thank you for proving me right by ending the conversation when i did. i've stated earlier in this exchange that stochastic processes, elements of stochastic calculus, and also probability theory, so clearly you've shown limited capacity to absorb outside arguments that collide with your world view. i expected that. but for you to state things i've already stated, as if they now reinforce your point, is ridiculous and demonstrative of your ignorance. mathematical finance is not pure mathematics.
you do realise the great sir andrew wiles has rejected mathematical finance, right? http://www.ibtimes.co.uk/andrew-wiles-maths-oxford-university-bankers-times-511492
the eradication of the mathematical finance page would be a good thing, as it has done nothing but harm to [[stochastic processes] and other fields. sir wiles' critiques quite rightly make this point verbose. he is a pure mathematician and he has rejected you guys, do you understand that?
further, i clearly asked you for a measurable function example, with 100 time points, and instead you go to a book of dr sinai's and give me a definition of measurable function. there is nothing in your definition that rejects what i'm saying (skip the next two lines if you want this explanation now).
since the "measured" values in your field are generated by a computer or human (how many times do i have to say this), the space is finite. you are not measuring nature!! a price is some arbitrary value assigned to the asset/security, and either way: these values (at one point or another) originated from a human!!
now, if i assume an asset or security's prices over time is measurable, then i should be able to plot it as a function of time, where f(t) would represent the asset/security's value at time t.
it says that the probability measure over the entire space is 1. it is essentially saying that the sum of probabilities must equal 1. the same must hold for a lebesgue measure, except since we start with a measurable function over time (see last sentence from above), we must first calculate the area between the arbitrary function (like the one shown in measurable function on the right hand side) and the x axis. the measure which best captures contour line represented by the function over the time interval can be calculated by constraining the area to equal value 1.
in fact, the area value doesn't matter much because the value acts as a constraint, and a higher number doesn't increase the cardinality of the set, which, according to sigma-algebra#definition, is 2^64 (unsigned). i can't speak much more about the calculation as it's not "go time" yet.
i then related the example of calculating area in terms of the riemann integral, Riemann-Stieltjes integral), and how the result of these integrals is a SUMMATION of an INFINITE number of RECTANGLES along the x axis. looking at the right hand side of measurable function, notice how there are areas being taken? this is common to all three integrals (taking the area under the curve), which is absent in your field.
again, we see geometry play an important role for all three integrals in calculus and i've yet to see anything from you similar to this. did you not see the plot on the right hand side of measurable function is showing the measurement of areas under the contour represented by the function? that's not just for lebesgue measures. it's for any function plotted against time (such as an asset or security price's fluctuations). you do realise that the red shaded area is similar to how one can view the trapezoid rule and rectangle rule, yes? so again, there is nothing stopping your field from demonstrating the rigorous properties of its integral (which would greatly improve its exposure and popularity) except the fact that it doesn't act uniformly across the entire real line, as the lebesgue measure does (because it is THE probability measure)
i do not understand why you're linking the normal distribution. i won't ask though, because i think it's cute you think your latex makes you look sophisticated. i find it especially comical because you're mentioning borel sets, which implicitly use topology, which, in turn, uses geometry! this geometry is something you've refused to acknowledge or demonstrate for any measurable function (again, i said plot a function over 100 time points and let that function represent the asset price over time. third time i've asked.).
lastly, you are aware that dimension is a page, right? and that on that page, you'll see how 'point' is defined is not as clear as you've made it out to be. in fact, it is MY area that determines when a dimension is a point, thank you very much lol. and a point in time" on the world line is one dimension. i.e. we are three dimensional bodies that exist over time, and a time point is an instant of time at which we can record measurement (like the purported values of the asset prices, or the neuronal mass activity measured by fMRI).
in closing: it seems you think you're smart, when i've seen little evidence to suggest otherwise. i suspect you are obligated to continue your charade to convince your financial victims that you know what you are talking about, when, unfortunately, you do not. the fact you're a PhD in "financial engineering" and thinking mathematical finance is pure mathematics is disappointing but unsurprising; especially when sir andrew wiles, arguably the greatest living mathematician alive, has rightfully criticised your field. clearly our differences lie solely on the importance of the geometric intuitions that are central to vanilla calculus, but conspicuously absent in mathematical finance, and which you think is acceptable. a few lines of latex and you're acting like you know something. so be it, lol. later.
can't wait to read your response in defense i know you need the "last word" so you can sleep thinking "i beat that guy, i'm smart". time will ultimately determine if your perception of the entire situation is correct, regardless of your "last word". g'day.

@Zfeinst: Please don't feed the troll. McKay (talk) 04:48, 4 April 2016 (UTC)