Consider a probability measure $\mu(\cdot)$ on $\mathbb{R}^2$ and another probability measure $S(\cdot)$ on the unit sphere $\mathbb{S}_1$.
From a polar coordinate standpoint, assume for any set $\Lambda\subset\mathbb{S}_1$ such that $S(\partial\Lambda) = 0$, it holds $$\mathbb{P}(\Theta\in\Lambda|R\geq r) = \frac{\mu(\{R\geq r, \Theta\in\Lambda\})}{\mu(\{R\geq r\})}\rightarrow S(\Lambda)\text{ as } r\rightarrow\infty$$
My question is: in what sense could I define $$\mathbb{P}(\Theta\in\Lambda|R= r)$$ and if this term is well-defined, then as the above assumption, could we show $$\mathbb{P}(\Theta\in\Lambda|R= r)\rightarrow S(\Lambda)$$ weakly as $r\rightarrow\infty$ (This statement is so intuitively convincing to me)
Thank you