Measure Theory/Lexicon

Lexicon

Throughout this course I use standard symbols and words, as well as rarer symbols and words, as well as some symbols and words which I have invented because I hope they help the learning process. Here I will try to catalog everything that is either rare, or made up.

Words

• interval

I define this in a slightly unusual way. I define an interval as any set of real numbers which is connected, using the standard topology. This means that both singleton sets, as well as the empty set, count as intervals. If we like to see this in interval notation, we could say that it is because $[a,a]$ is written in interval notation, and is the same thing as the singleton set $\{a\}$. Also $(a,a)$ is written in interval notation and it is the empty set.

• length-measure

Almost universally referred to as the Lebesgue measure, it is the infimum over all over-estimates.

• open interval over-approximation

For a set $A\subseteq {\mathbb {R}}$ I call any collection of open intervals ${\mathcal {O}}$ an open interval cover of A if $A\subseteq \bigcup _{I\in {\mathcal {O}}}I$.

• over-estimate

For a given open interval over-approximation, ${\mathfrak {O}}$, I define the corresponding over-estimate to be the sum over all lengths of intervals inside.

$e=\sum _{I\in {\mathfrak {O}}}\ell (I)$

• step function

Any linear combination of indicator functions, where the indicator functions are for intervals of real numbers.

Symbols

• $\smallsetminus$

My preferred symbol for set-minus. $A\smallsetminus B$ is equivalent to $A\cap B^{c}$.

• $\Delta$

The symmetric difference. $A\Delta B$ is equivalent to $(A\smallsetminus B)\sqcup (B\smallsetminus A)$.

• $\sqcup$

The "disjoint union" of two sets. It is the same thing as a union, but when the sets are disjoint we use this symbol to further express that the sets are disjoint.

• $\lambda$

The symbol for the length measure.

• $\lambda ^{*}$

The symbol for the length outer-measure.