I recently started learning measure theory but had a pretty hard time finding real-world examples of the different measures. So this is a list of real-world examples of some different, common measures.

The count measure

If I have 4 apples, their count measure is 4. The count measure is just normal counting.

The Dirac measure

You are camping in a forest with some trees. You've set up your tent and it was a lot of work, so you're not going to move. It's daytime. The trees cast shadows, but the tree cover is sparse enough that there's still some sunlight that hits the ground. As the day goes on, the shadows move, at times covering your camp, at times leaving you in the sun. The dirac measure for whether a shadow is on your tent is something that outputs 1 when a shadow is on your tent, and 0 when there isn't a shadow on your tent.

Imagine the shadows (green circles) changing position over the course of a day. The point marked 'you' stays where it is. Whenever a green circle overlaps the red dot, the dirac measure is 1; whenever not, it's 0.

Formally, if $δ_{x}$ means "Tent is in shadow", and A is the collection of shadow regions, the mathematical expression of this dirac measure is $δ_{x} (A) = {\begin{matrix} 0, x \notin A 1, x \in A \end{matrix} .$

The dirac measure always looks like this, whatever meanings you attach to $δ_{x}$ and the set A.

(Since x is a point, the analogy breaks down a little: shadows are not allowed to partially overlap the tent. They cover it fully, or they don't touch it. We could fix this above by talking about "the northwesternmost atom of fabric of the tent" instead of "the tent").

(The dirac measure is similar to the indicator function (not measure!) $1_{A} (x)$ , which takes a point $x$ and returns whether or not it's in A. For example, if you were walking around the forest instead of entented in it - that is, if the red point in the picture was moving around and the shadows didn't move, instead of the other way around - you could set up an indicator function $1_{a} (x)$ to be 1 when you're inside a shadow, and 0 when you're outside one. A previous version of this post confused the dirac measure with the indicator function; the difference is that $δ_{x}$ takes a set A as input and assumes $x$ fixed, while $1_{A}$ takes a point $x$ as input and assumes the set A fixed. Thanks to deluks917 in the comments for the correction.)

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Measure Theory