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 $cos$ $L^1$?
Best Answer
Given a measurable space $X$ equipped with a measure $\mu$, a function $f : X \to \mathbb{C}$ which is defined almost everywhere (that is, up to a set of measure $0$) is said to be an element of $L^1$ if
$$\int_X |f| d\mu < \infty$$
More properly, we have to identify functions which are equal almost everywhere, so the elements of the Lebesgue space $L^1$ are really equivalence classes of functions under the relation of being almost everywhere equal - but this is a technical note.
Practically speaking, a real or complex valued measurable function on the real line with respect to Lebesgue measure is an element of $L^1$ if $$\int_{-\infty}^{\infty} |f(x)| dx < \infty$$ So a function like $\chi_{[0,1]}$ which is $1$ on $[0,1]$ and $0$ outside is in $L^1$, as is $e^{-x^2}$.
If $f$ is continuous enough, this coincides with the usual Riemann integral. Now it's fairly easy to prove that $$\int_{\mathbb{R}} |\sin x| dx$$ can't be finite, so $\sin x \notin L^1(\mathbb{R})$. In some sense, $L^1$ functions have to decay to $0$ at $\pm \infty$: In fact, one way to think of $L^1$ is that it's the completion of
$$C_C = \{\text{continuous functions supported on a compact set}\}$$
under the metric induced by integration (again, with slight technical caveats).
So in short, ignoring the technical definitions, $L^1$ functions are exactly those functions which have small enough spikes and decay fast enough that $\int |f| < \infty$.