Real Analysis – Weak Convergence of Characteristic Functions Sequence

functional-analysislp-spacesreal-analysis

I am trying to produce a sequence of sets $A_n \subseteq [0,1] $ such that their characteristic functions $\chi_{A_n}$ converge weakly in $L^2[0,1]$ to $\frac{1}{2}\chi_{[0,1]}$.

The sequence of sets
$$A_n = \bigcup\limits_{k=0}^{2^{n-1} – 1} \left[ \frac{2k}{2^n}, \frac{2k+1}{2^n} \right]$$
seems like it should work to me, as their characteristic functions look like they will "average out" to $\frac{1}{2} \chi_{[0,1]}$ as needed. However, I'm having trouble completing the actual computation.

Let $g \in L^2[0,1]$, then we'd like to show that
$$
\lim_{n \to \infty} \int_{[0,1]} \chi_{A_n} g(x) dx = \int_{[0,1]} \frac{1}{2}\chi_{[0,1]} g(x) dx = \frac{1}{2} \int_{[0,1]} g(x) dx
$$
We have that
$$
\int_{[0,1]} \chi_{A_n} g(x) dx = \sum\limits_{k=0}^{2^{n-1}-1} \int_{\left[ \frac{2k}{2^n}, \frac{2k+1}{2^n} \right] } \chi_{A_n} g(x) dx
$$
Now I am stuck, as I don't see how to use a limit argument to show that this goes to the desired limit as $ n \to \infty$. Does anyone have any suggestions on how to proceed? Any help is appreciated! 🙂

Best Answer

  1. The sequence $\{\chi_{A_n}\}$ is bounded in $L^2$, and the polynomials are dense in $L^2[0,1]$, so it's enough to consider the case $g$ polynomial.
  2. By linearity, the case $g(x)=x^p$, $p\in\Bbb N$ is sufficient.
  3. The involved integral are explicitely computable. Then write $$(2k+1)^{p+1}-(2k)^{p+1}=\sum_{l=0}^p\binom{p+1}l(2k)^l,$$ and consider the term $l=p$ separately.
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