Properties That Hold Almost Everywhere

# Properties That Hold Almost Everywhere

Recall from awhile back on the Subsets of Real Numbers with Measure Zero page that a subset $S \subset \mathbb{R}$] is said to have measure $0$ denoted $m(S) = 0$ if for all $\epsilon > 0$ there exists a countable open interval covering $\{ I_k = (a_k, b_k) : k \in K \}$ ($K$ is a countable indexing set), $\displaystyle{S \subseteq \bigcup_{k \in K} I_k}$ and such that:

(1)
\begin{align} \quad \sum_{k \in K} l(I_k) = \sum_{k \in K} (b_k - a_k) < \epsilon \end{align}

Before we move on further with getting closer to discussing Lebesgue integrals we will first need to describe a very special type of property which we define below.

 Definition: Let $D \subseteq \mathbb{R}$. A property is said to hold Almost Everywhere on $D$ if there exists a subset $S \subseteq D$ whose measure is $0$, i.e., $m(S) = 0$ and such that the property holds on all of $D \setminus S$.

Sometimes the abbreviation "a.e." is used to acknowledge that a property holds almost everywhere.

In other words, a property holds almost everywhere on a set $D$ if the only points in which it does not hold constitute a set of measure $0$.

For example, consider the following function:

(2)
\begin{align} \left\{\begin{matrix} 0 & \mathrm{if} \: x \in \mathbb{R} \setminus \mathbb{Q}\\ 1 & \mathrm{if} \: x \in \mathbb{Q} \end{matrix}\right. \end{align}

Since $\mathbb{Q}$ is a countable subset of $\mathbb{R}$ then we can say that the function $f$ equals to $0$ almost everywhere on $\mathbb{R}$.

For another example, consider the following function:

(3)
\begin{align} \left\{\begin{matrix} x & \mathrm{if} \: x \in \mathbb{R} \setminus \mathbb{Q}\\ 0 & \mathrm{if} \: x \in \mathbb{Q} \end{matrix}\right. \end{align}

In this particular example, we can say that $f$ is strictly increasing almost everywhere on $\mathbb{R}$.