What does it mean for something to be $L^2(G)$ for compact G For the particular example I'm interested in, I know a particular field F (maybe scalar, vector or tensorial) "lives in"  $L^2(SU(2))$. What exactly does that mean?
My intuition says that $$F\epsilon SU(2)$$   Or rather that the field consists of elements of $SU(2)$. Is that right?
ELABORATION
Based on the comments below, I think I need to elaborate on my question. First of all I am dealing strictly with compact groups. I was recently reading about the Peter-Weyl theorem, which for a simplified version of my case reads:
$$L^{2}(S^{3})=L^{2}(SU(2))$$ I understand that this ultimately is a generalization of the Fourier series expansion in that we can say that the $mth$ representation of $SU(2)$ is equal to the $mth$ harmonic in a series expansion of some scalar/vector/tensor on $S^3$. Thus we can say that:
$$F=\sum_{m}^{\infty}a_{m}SU(2)^{m}$$
Where we are denoting $SU(2)^m$ to be the $mth$ representation of $SU(2)$ and $a_m$ is the equivalent of the fourier coefficient. If $F$ is composed of a sum of irreducible representations of $SU(2)$ then isn't $F$ itself act as an element of $SU(2)$? Please excuse any poor notation/terminology, I'm quite new to Group theory, but definitely want to learn.
Apparently there is still some ambiguity in my question as per the comments below. In a paper describing The harmonics on $S^3$ they write that:

the
  expansion coefficients V
  are simply the integral pro-
  jections of the vector field onto the corresponding vector
  harmonics

 A: Compact groups such as $SU(2)$ have a unique normalized Haar measure $\mu$ with total measure $1$. Haar measure on a compact group $G$ allows you to define the Hilbert space called $L^2(G)$ consisting of (equivalence classes of) measurable square-integrable functions 
$$f : G \to \mathbb{C}$$
where square-integrable means that the integral $\int |f|^2 \, d \mu$ converges, where $d \mu$ refers to Lebesgue integration with respect to Haar measure, and the equivalence relation is that $f \sim g$ if $\int |f - g|^2 \, d \mu = 0$. So elements of $L^2(G)$ are (equivalence classes of) functions from $G$ to $\mathbb{C}$, not elements of $G$. 
Due to the invariance of Haar measure, $G$ naturally acts on $L^2(G)$, making it a linear representation of $G$, and one piece of the Peter-Weyl theorem says that this representation decomposes as a Hilbert space direct sum of $\dim V$ copies of every irreducible representation $V$ of $G$. 
The representation theory of $SU(2)$ itself is extremely well-understood; it has exactly one irreducible representation $V_d$ of each dimension $d$, which is isomorphic to the representation of $SU(2)$ on homogeneous polynomials in $2$ complex variables of degree $d - 1$, and so the decomposition of an element $f \in L^2(SU(2))$ in terms of these irreducibles consists of, for each positive integer $d$, a collection of $d^2$ scalars describing the component of $f$ living in the sum of $d$ copies of $V_d$. This data can also be thought of as a collection of $d$ homogeneous polynomials in $2$ complex variables of degree $d - 1$. 
