# Lebesgue Measure The Lebesgue Measure informally is simply the "length" of a set of real numbers. ### Definition Consider the following example. Let us have a function $f$, and we integrate this function from $a$ to $b$: $\int_a^b f(x) dx$ Now, say that we have some set $x$ that lies inside the closed interval from $a$ to $b$: $x \subseteq [a, b]$ $x$ does not need to be an interval itself, it can be anything inside of the interval $[a,b]$ (note: technically $x$ cannot be *anything*, for more see the axiom of choice). The lesbegue measure of $x$, $\mathcal{L}(x)$, will determine how much the set $x$ contributes to our integral. We can now define the Lesbegue Measure as follows: > Definition: Consider the interval $(a,b)$. $\mathcal{L}\big( (a,b) \big) = b - a$. > > Let $X \subseteq \mathbb{R}$. Now let us define a family of open intervals that are indexed by the natural numbers: > $I_1 \cup I_2 \cup I_3 \cup \dots$\ > If > $X \subseteq I_1 \cup I_2 \cup I_3 \cup \dots$ > then we will say that this family of open intervals, $\{ I_n \}_{n \in \mathbb{N}}$, **covers** $X$. > > For the these intervals $I_n$, we are going to take their lesbegue measure, and then we are going to sum them: > $\sum_{n=1}^{\infty} \mathcal{L}(I_n)$ > But, at this point the above sum may severely *over estimate* the lesbegue measure of $X$ (since $X$ can be a tiny fraction of the union of our intervals). To fix this, we will look at *all* covers: > $\big\{ \sum_{n=1}^{\infty} \mathcal{L}(I_n) \big| \{ I_n \}_{n \in \mathbb{N}} \text{ covers } X \big\}$ > And then take the *infinum* of that set, resulting in our **Lebesgue Measure**: > $\mathcal{L}(X) = inf\big\{ \sum_{n=1}^{\infty} \mathcal{L}(I_n) \big| \{ I_n \}_{n \in \mathbb{N}} \text{ covers } X \big\}$ ### Examples Note that the natural numbers and the integers both have a lebesgue measure of 0 (see more [here](https://youtu.be/0VD3BWDLmU0?t=530)). --- Date: 20210714 Links to: [Mathematics MOC](Mathematics%20MOC.md) Tags: #review References: * [Comparing the sizes of sets in different ways](https://www.youtube.com/watch?v=0VD3BWDLmU0)