TITLE: What does it mean to be an $L^1$ function?
QUESTION [21 upvotes]: I am struggling to understand what the space $L^1$ is, and what it means for a function to be $L^1$.
A friend told me that a function $f$ is $L^1$ if $\int_\mathbb{R} |f|$ is finite.
It is $L^2$ if $(\int_\mathbb{R} |f|^2)^{1/2}$
Firstly is this correct?
I have looked online but the definitions seem complicated. I'm not studying a Lebesgue integration course, and I just want a basic understanding of what these spaces are.
He explained the spaces as saying they contained functions that decay to zero, is this correct?
Could somebody provide a simple definition/intuitive explanation for me to view these spaces, and give me some examples of functions they contain(and better yet functions they don't contain).
I have never studied Lebesgue Integration, and am now taking a graduate Fourier Analysis course, so perhaps I should study it in some more detail, but for the moment I'd just like to understand a bit more about these spaces. 
-----EDIT-------
So for example are $sin$ and $cos$ $L^1$?

REPLY [2 votes]: We can, in fact, give a definition that doesn't require the Lebesgue measure or measure theory. We define a function $f: \mathbb{R} \to \mathbb{R}$ to be Lebesgue integrable ($f \in \mathcal{L}^1(\mathbb{R})$) iff there is a sequence $(f_n)_{n=1}^\infty$ of continuous, compactly supported functions $f_n \in C_c(\mathbb{R})$ such that the following two properties hold:

$\displaystyle \sum_{n=1}^\infty \int_{\mathbb{R}} |f_n| < \infty$,
If $x \in \mathbb{R}$ satisfies $\displaystyle \sum_{n=1}^\infty |f_n(x)| < \infty$, then $ \displaystyle\sum_{n=1}^\infty f_n(x) = f(x)$.

In the above, the integral $\int_{\mathbb{R}} |f_n|$ is the Riemann integral; it exists and is finite for all $f_n$ because $f_n \in C_c(\mathbb{R})$.
Here I've written $\mathcal{L}^1(\mathbb{R})$ instead of $L^1(\mathbb{R})$ because they're technically different. The former contains the Lebesgue integrable functions, and the latter treats Lebesgue integrable functions which are equal almost everywhere as equivalent.
The upshot is that in some sense, we can think of a function $f \in \mathcal{L}^1(\mathbb{R})$ as the limit of $C_c(\mathbb{R})$ functions which vanish at infinity in a way that restricts their (absolute) integrals over all of $\mathbb{R}$ to be finite, and hence $\int_\mathbb{R} |f|$ to be finite.