202405211626
Status: #idea
Tags: Measure Theory

Lebesgue Measures

A Lebesgue measure is a type of measure defined on a continuous sample space $\Omega$ and more specifically on the Borel set $\mathcal{B}$ associated with that sample space, effectively giving us a concept of "length", "area" or "volume" thus a "measure" over that topological space.

Note the first usage of measure is the idea of a function from $\sigma -$algebra to $[0,\mathrm{\infty }]$ the second use of measure (between quotation marks) represents the idea of intuitive notion measure used in the real world.

Properties of Lebesgue Measures

The motivation is that we wanted to define measures on the reals $\mathbb{R}$ and whatnot in a rigorous way. For those measures, for those "standardization of volume" to make sense, we require these two properties:

• $\mu ([a,b])=b-a,b>a$
• $\mu (x+A)=\mu (A)$, for all $A\in 2^{\mathbb{R}}$ and $x\in \mathbb{R}$

The first property tells us that the Lebesgue measure gives us the length of an interval for any arbitrary interval $[a,b]$.
The second property tells us that $\mu ([10,12])=\mu (0,2)=\mu (2,4)$ because while the numbers are different they all have the same length. The length of the measure is invariant no matter where it is taken.

The catch is that We Cannot Define a Lebesgue Measure on Power Set of a Uncountably Infinite Sample Space. Which stumped measure theorists for a while, this in fact lead to the invention of a whole new measure theory centralized around the idea of Borel-Sigma Algebras.

Why Can't Lebesgue Measures be defined on the whole power set $2^{\Omega }$?

Well it's something you have to prove.
The proof is shown here: Not Everything Is Lebesgue Measurable

My attempt at an explanation: Why Is Everything Not Lebesgue Measurable?

The proof makes use of a few concepts explained here:
Axiom Of Choice

Usage in Probability Theory

By using the standard and more easily manipulable concept of algebras we can surgically select elements of $2^{\Omega }$ that are of interest to us and assign them probabilities. These probabilities since assigned on algebras that are not $\sigma$ can hardly be called probability measures and are typically denoted $P_0$. For the uniform case for example our pseudo-measure $P_0$ is for an arbitrary algebra $\Xi$ is:

to which we add the specification that:

This is well and all, except that probabilities do not work well on general algebras more specifically, the concept of closure under countable unions is a crucial one to allow any meaningful analysis of probabilities.

This is where we stand on the shoulders of giants and invoke the Caratheodarry's Extension Theorem by which we know that if our pseudo-measure

• assigned a probability to $\Omega$ of $1$
• and was countably additive (for the cases where the countable union actually is in the algebra)
  then there exists an actual measure that agrees with our pseudo-measure for all the measures, and not only that, that it extends to a $\sigma -$algebra (which typically will be the $\mathcal{B}$).

Yes, you've guessed it: that measure is called the Lebesgue measure.

Note that the Lebesgue measures are by no means restricted to probability spaces and are a really important concept in Measure Theory. But for Probability Theory which is not much more than a special case of the latter, they play a pivotal role in allowing us to analyze Continuous Probability Spaces.

Note, that based on the above we can show that for any singleton set element of our $\sigma -$algebra the probability will be $0$.

Weird Stuff and Considerations

What is the probability of $P(\mathbb{Q}\cap \Omega )$.
This is really weird, but while there is an infinity of rational numbers $\mathbb{Q}$ in $\Omega$, we know that all singleton element has probability $0$. Since all these singleton elements by definition contain only one element, and they are all distinct, it follows that all of those probability measures are individually equal to $0$. Then by countable additivity we have $0$.

Thus as counter-intuitive as this might seem, if I throw a dart and I have a uniform chance of reaching any individual number on that then the probability of my number landing on a rational number is mathematically $0$.

More then that, it is not approximately $0$, or $0$ in some sense--it is exactly $0$.

This leads to the observation that $P(A)=0$ does not mean that $A$ never occurs, it simply means that its probability is $0$.

(Observe that since $P(\mathbb{Q}\cap \Omega )=1$ since irrational numbers are the complement of rational numbers, it follows that $P([\mathbb{R}\setminus \mathbb{Q}]\cap \Omega )=1)$ even though both sets are non-empty. The intuition is that if I throw a dart, I am almost guaranteed to land on a irrational number, and it really unlikely that I will land on a rational number. But the former COULD occur!)

In a Continuous Probability Spaces an event with a probability of $0$ could still occur, but it is not expected to occur--not likely to occur in a probabilistic sense. Note that this notion of "expected to occur" is unrelated to the probabilistic notion of Expected Value.

Observe that the weirdness goes further, while $P(\Omega )$ has a probability of $1$, for literally any of the uncountably-many-individual elements in the interval its probability is $0$. This is a reflection of how in Continuous Probability Spaces, probability measures "length" which is then likened to an esoteric sense of likeliness.

This is another example of the disconnect between everyday language and mathematics.

This is a feature and not a bug of Continuous Probability Spaces, it is with humour that we say that in continuous probability spaces $0$-probability events occur all the time, in a sense (at the highest level of resolution) they are the only thing that can occur.

Spoiler, the only impossible event is $\varnothing$. Informally, an Impossible Event is an event that does not contain any of the possible outcomes of a random experiment.

Refer to this for a more in-depth comparison: Impossible Event vs 0-Probability Events

This tells us two things

• Be careful of "real-world intuition" the deeper you go into mathematics
• Infinity does not in any way guarantee a positive measure, in fact The Cantor Set has A Probability Measure of 0. This is meaningful because the Cantor Set is not only infinite, it is uncountably infinite.

Relevant Links