I guess that amounts to if there is a continuous $f$ with $\mathbb{P}(\mathbb{L}_{f(\alpha)}) \cap \mathbb{L} = \mathbb{L}_{f(\alpha+1)}$
I seem to remember reading that it is, but I forget where or when I read it, why it's true, what f is, and I can't find it now.
Thanks for any info.