Hey guys!

I'm working through a proof from an old paper for my project and I'm really stumped with this notation. I want to understand the meaning of:

For $\displaystyle x$, $\displaystyle y \in S^{2}$, $\displaystyle k \ne l$

$\displaystyle \int_{S^2} P_k({x}{z})P_l({y}{z}) dz=0$

Where the $\displaystyle P_k$ are Legendre polynomials degree $\displaystyle k$.

Now I know that this is a kind of more generalised orthogonality relation and the legendre polynomials take scalar arguments (at least I think I know that...) in which case I think it would mean

For $\displaystyle \boldsymbol{x}$, $\displaystyle \boldsymbol{y} \in S^{2}$, $\displaystyle k \ne l$

$\displaystyle \int_{S^2} P_k(\boldsymbol{x}\cdot \boldsymbol{z})P_l(\boldsymbol{y} \cdot \boldsymbol{z}) d\boldsymbol{z}=0$

But then I dont understand the meaning of $\displaystyle d\boldsymbol{z}$.

Ive read in an old text that used the notation "for $\displaystyle x=(x_1,x_2, \ldots, x_n) ... $where $\displaystyle dx=dx_1 dx_2 \cdots dx_n$ is the ordinary Lebesque measure", but I haven't been taught the Lebesque stuff and it looks real tricky...

So if anyone could shed a little light on whats going on, or point me to somewhere that does if its elementary (and im just being stupid) then id be so grateful!