Essentially bounded function on $\mathbb{R}$

I don't really know a lot about measure (just finishing my undergrad) so I'm not really on good terms with this. So, let $L^\infty[a,b]$ denote the space of all essentially bounded functions on $[a,b]$ with the norm $\left\| f \right\|_\infty = \operatorname{ess} \sup_{x\in[a,b]} |f(x)|$. What would exactly be the difference between a bounded and an essentially bounded function (also, the difference between the supremum and essential supremum)? If I'm not actually dealing with measures, but just with real variable functions $f:[a,b]\to\mathbb{R}$, can I omit the "essentially" part and just say that $L^\infty[a,b]$ denotes the space of bounded functions, and, additionally, omit the $\text{ess}$ in the definition of the norm? I understand that this seems like a silly question, but I've tried Googleing this and found nothing useful.

Solution 1:

An essentially bounded function $f\in L^{\infty}([a,b])$ is explicitly related to the idea of measure, so I don't think there's a way to understand this without measures. The definition can be stated that $f\in L^{\infty}([a,b])$ if there is a $g$ measurable on $[a,b], f=g$ except on a set of measure zero, and $g$ is bounded.

Two examples: $$f_1(x)=\begin{cases} x & \text{ if } x\in\mathbb{Q}\\ 0 & \text{ otherwise}\end{cases}$$Then $f_1=0$ except on a set of measure $0$, namely $\mathbb{Q}$, so $f_1$ is essentially bounded on $\mathbb{R}$, although $f_1$ is clearly not bounded on $\mathbb{R}$.

Next,

$$f_2(x)=\begin{cases} 1 & \text{ if } x\in \mathbb{Q}\\ 0 & \text{ otherwise}\end{cases}$$ Then $f_2=0$ except on a set of measure zero, so $f_2$ is also essentially bounded. Moreover, $f_2$ is also bounded, $|f_2|\leq 1$ and $\sup |f_2|=\|f_2\|=1$. However, note that $\|f_2\|_\infty\not=\|f_2\|$. So even if a function is essentially bounded and bounded, its essential bound is not necessarily equal to the bound: it can be greater or less than the bound. Thus, the essential bound and the supremum bound are fundamentally different.

Solution 2:

This is just a byproduct of the fact that you are dealing with equivalence classes of functions in $L^\infty$, not individual functions. In measure theory, you assume that all functions that are the same apart from a set of measure zero are equal, since their integrals are the same and behave the same way for anything you care about in measure theory. So an essentially bounded function is a function that is equivalent to a bounded function. for example, a function that outputs $f(x)=x$ at $x \in \mathbb{Z}$ but $f(x)=0$ for all $x \in \mathbb{R}\setminus \mathbb{Z}$ is unbounded, but clearly is equivalent to the zero function from the perspective of the lebsegue measure.