Are $L^1\cap C$ Functions Essentially Bounded?

Question

Let $f:\mathbb R\to\mathbb C$ be continuous, and let the Lebesgue integral

$$\int_\mathbb R\left|f\left(x\right)\right|\,dm\left(x\right)<\infty.$$

I am trying to show that $\text{ess}\sup\left|f\right|<\infty$, i.e.,

$$\inf\left\{a:a\in\mathbb R\text{ and }m\left(\left\{x\in\mathbb R:\left|f\left(x\right)\right|>a\right\}\right)=0\right\}<\infty.$$

Is this even true?

Answer 1

It's not true. Think of a nonnegative function $f$ with bumps at each point $x=1,2, 3,\ldots$. The bumps get taller and narrower, so that the sum of the areas is finite, but the heights grow without bound.

Answer 2

Consider a function $f$ such that the graph of $f$ restricted to $[n,n+1]$ is a triangular spike of height $n$ and width $1/n^3$. Then $f$ gets arbitrarily large on sets of positive measure but the $L_1$ norm is $\frac12\sum_1^\infty \frac1{n^2} < \infty$.

Answer 3

It is not true as the standard counter example communicated by Umberto P. and Jamie Radcliffe shows.

The in quotation marks "right" way to think about the necessary decay of $L^1$ functions is that they on average tend to zero; meaning that for all $\epsilon>0$:

$$ \lim_{a\to\infty} \int_{a-\epsilon}^{a+\epsilon}|f(x)|dx = 0$$

So although $f$ is itself not necessarily essentially bounded, its local average is.