For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$, $$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$ We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by $$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$ Let $f = \lim_{n \rightarrow \infty} \psi_{n}$. Show that $f$ has compact support and that if $\psi_{0}$ is positive and not identically zero, then so is $f$.
For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Let $f = \lim_{n \rightarrow \infty} \psi_{n}$. Show that $f$ has compact support and that if $\psi_{0}$ is positive and not identically zero, then so is $f$.