So why is this only “almost surely” a math blog? For technical reasons, it’s because maybe in the future some posts will not be about math but related subjects (or unrelated, perhaps in 2026 this blog will be about pistachios, who knows honestly). But the major reason I chose this title is as a reference to the use of the phrase in probability (and also measure theory etc.).
When we say an event happens “almost surely” in probability, we mean it happens with probability equal to one. Similarly, we say an event happens “almost never” if it happens with probability zero. These are pretty fitting names for the definitions, however the use of the word “almost” can lead to some pretty funny implications in my view.
Let’s take an example. Suppose we have a simple (finite) probability space, say a magician picking from three balls in a hat uniformly, and denote it by Ω:
So from the way we defined the space, with probability one the magician picks out a ball, so almost surely he picks out a ball. Additionally, with probability zero he picks out, say, an elephant. So he almost never picks out an elephant.
Here’s the thing, the use of almost in the definition may give a hint (if we stop thinking of the definitions mathematically but rather directly from their name) that there is a chance the magician doesn’t pick out a ball, but an elephant. Maybe, just maybe in a fringe chance he will pick out an elephant (if we ignore the impossibility of the situation mathematically and the fact most people can’t pick up an elephant with their bare hands)
Well, actually it is mathematically accurate to say the magician will never pick out an elephant. The thing is, the way we defined the probability space didn’t account for the event. We need to add the event of the magician picking an elephant and give it probability zero for it to be true that the magician almost never. but not never picks out an elephant.
I guess the reason I find the definition a bit humorous is due to it being very abundant in probability. So much so that it makes you wonder sometimes about what will happen when the “almost always” won’t occur.
Some other definitions with funny implications:
In topology, there is a group of properties for topological spaces called separation axioms. These include:
- Normal topological space
- Completely normal topological space
- Perfectly normal topological space
It feels like these definitions form a scale of assuring someone there is nothing wrong with your topological space. It doesn’t bite, it isn’t radioactive, it’s hopefully not incredibly ugly, it’s just a perfectly, normal topological space.
Large and Huge cardinals:
In the subject of set theory, which deals in particular with different “sizes” of infinities called cardinals, there are definitions of ‘Large’ and ‘Huge’ cardinals (more precisely, large cardinal property). It feels a bit odd describing a cardinal as large. Like, it is infinite, of course it is going to be large why won’t it be. These definitions, however, make much more sense in the context of set theory.
I think this one speaks for itself…