Let $\{T_k\}$ be a dyadic decomposition of $\mathbb{R}$, namely, the collection of dyadic intervals of length $2^{-k}$, $k\in\mathbb{N}$. Consider the averaging operator $$E_k(f)=\sum_{J\in T_k}\left(\frac{1}{|J|}\int_J f d x\right) \chi_{J},$$ where $f$ is nonnegative and $f\in L_{loc}^1{(\mathbb{R})}$.
Do we have $E_{k}(f)\to f$ as $k\to +\infty$?
(If necessary, we can let $f\in A_{\infty}$, where $A_{\infty}$ is the Muckenhoupt weight)
I guess and hope it is right, any help will be appreciated!