Ergodic and Invariant measure on $\ell^p$ for backshift operator

Question

Define the weighted backshift operator on $\ell^p$ by $$ B_2:(x_1,x_2,\dots)\mapsto (2x_2,2x_3,\dots). $$ Is there an explicit example of a $B_2$-invariant probability measure on $\ell^p$ for which $B_2$ is erogidic?

Answer 1

Consider eg. any fixed point of the shift such as the sequnce $(2^{-n})$ and concentrate the measure on it, ie. take the Dirac measure at this point. It's invariant and ergodic.