How to intuitively visualize measure theory concepts via diagrams? Visualizing Measure Theory via Diagrams / Drawings / Geometry
Hi,
I'm starting to learn about Measure Theory in order to study more advanced topics in Data Science. However, the explanations are always very Mathsy and have little to no geometrical intuition or understanding. When I say geometrical, I mean diagrams, images, drawings that help you visualize Measure Theoretic concepts in an intuitive way.
EDIT: I'm happy with diagrams that represent finite measure theoretic concepts!
The concepts that I'm trying to visualize are:

*

*Sample Space

*Event Space

*Probability Measure

*Probability Space

*Random Variable

*Probability Distribution

*Probability Density Function

*Markov Transition Kernel

Any help? I can give an example for the sample space and the event space:
Sample Space
Suppose I have a sample space $\Omega = \{\omega_1, \omega_2, \omega_3\}$ then I can visualize it as follows:

Event Space
Similarly, the event space is a Sigma algebra, which means it is non-empty, closed under complementation and closed under countable unions.

 A: TLDR;
I have written a blog post about it here.
VIsualizing Measure Theory Via Diagrams
I actually struggled myself so much with this. I feel like measure theory should be taught via intuition first, rather than heavy mathematics. I've tried to answer almost all of your measure-theoretic concepts, hopefully future answer will shed more light.
Sample Space
Your diagram for a sample space makes sense. If you have a sample space $\Omega = \{\omega_1, \omega_2, \omega_3\}$ then a simple diagram would look like a Ven Diagram:

Event Space
Again, as you said, an event space $\mathcal{F}$ is just a sigma-algebra of $\Omega$ so it will look something like this:

In the diagram above, I've copied yours but also have used colors to match complements. For instance, the complement of $\Omega$ is $\emptyset$, so they are both black.
Probability Measure and Probability Space
A probability measure maps the events between $[0, 1]$ as follows:

The triplet $(\Omega, \mathcal{F}, \mathbb{P})$ is called a probability space.
Random Variable
A random variable is a measurable function between measurable spaces. This means that given two measurable spaces $(\Omega, \mathcal{F})$ and $(\mathsf{E}, \mathcal{E})$ the pre-image of a set $B\in\mathcal{E}$ via the random variable $X$ is an element of $\mathcal{F}$, as shown below:

Probability Distribution
A probability distribution, or pushforward measure, is a way of assigning a measure to set in a sigma algebra for which we don't have a probability measure, by using a surrogate probability space.

Markov Kernel

Conclusion
Importantly, these diagrams are just for finite example and are quite simplified. However, in my mind, this is how I visualize measure-theoretic concepts when I can. Please do check my blog as I provide a bit more detail.
A: Sorry, I do not think that it's possible :)
Why do we need measure theory, $\sigma$-algebras, Borel sets? Because:

*

*simplest case: the sample space is finite, $\Omega=\{\omega_1,\dots,\omega_n\}$, there is a finite number of events (subsets in $\Omega$), $2^n$, every subset is "measurable", for example yon can just count its elements;

*hardest case: the sample space is uncountable, there are more subsets than real numbers, there are infinitely many nonmeasurable subsets. For example, if $\Omega=[0,1]$ and every $\omega$ (point) is equally likely, than $P(\omega)=0$, $P([0,a])=a$, where $a$ is the length of $[0,a]$, but there are subsets "without length", e.g. Vitali sets. See here.

This is why you need a $\sigma$-algebra of Borel sets (much smaller than the collection of all subsets of an uncountable sample space), because Borel sets are just open intervals, therefore measurable.
But uncountable sets are somewhat contrary to geometric intuition. For example, you can't draw a Venn diagram with as many points as there are in $[0,1]$, there are as many points in $[0,1]$ (a segment) as in $\{(x,y):x\in[0,1],y\in[0,1]\}$ (a rectangle).
My advice: try reading Shiryaev, Probability. In the first chapter he considers sample spaces whith a finite number of elementary events, algebras (not $\sigma$-algebras) which are (finite) collections of all the subsets of a sample space, their generation by decomposition of a sample space, then random variables, randow walk, martingales, Markov chains in this simple context. In the following chapters he extends what you have learned to uncountable sample spaces and $\sigma$-algebras.
I think that this is a lot easier than learning random variables etc. starting with $\sigma$-algebras and Borel sets.
