Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions on that measurable space and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)$ converge in mean to $f$ and $1_{A_n}$ to $1_A$ in measure. Show that then $f_n1_{A_n}$ converges in mean to $f1_A$.
Hello and good evening!
My idea is to show instead that $(f_n1_{A_n})_{n\in\mathbb{N}}$ converges to $f1_A$ in measure and that $(f_n1_{A_n})_{n\in\mathbb{N}}$ is uniformly integrable (because this together is equivalent to the convergence in mean of $(f_n1_{A_n})_{n\in\mathbb{N}}$ to $f1_A$).
I. Uniformly integrability
Consider any $\varepsilon>0$. $(f_n)_{n\in\mathbb{N}}$ is uniformly integrable (and converges to $f$ in measure what is needed below). So there exists an integrable function $h\geq 0$, so that
$$
\sup_{f_n\in (f_n)_{n\in\mathbb{N}}}\int 1_{\lvert f_n\rvert\geq h}\lvert f_n\rvert\, d\mu<\varepsilon.
$$
For this function $h$, it is
$$
(\lvert 1_{A_n}f_n\rvert-h)^+\leq 1_{\lvert 1_{A_n}f_n\rvert\geq h}\lvert 1_{A_n}f_n\rvert\leq 1_{\lvert f_n\rvert\geq h}\lvert f_n\rvert.
$$
So it follows that
$$
\int (\lvert 1_{A_n}f_n\rvert-h)^+\, d\mu\leq\int 1_{\lvert f_n\rvert\geq h}\lvert f_n\rvert\, d\mu\leq\sup_{f_n\in (f_n)_{n\in\mathbb{N}}}\int 1_{\lvert f_n\rvert\geq h}\lvert f_n\rvert\, d\mu<\varepsilon
$$
what means that
$$
\sup_{1_{A_n}f_n\in (1_{A_n}f_n)_{n\in\mathbb{N}}}\int (\lvert 1_{A_n}f_n\rvert-h)^+\, d\mu<\varepsilon.
$$
So $(1_{A_n}f_n)_{n\in\mathbb{N}}$ is uniformly integrable.
II. Convergence in measure
Consider any $\varepsilon>0$. It is
$$
\left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_Af\rvert>\varepsilon\right\}\\=\left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_{A_n}f+1_{A_n}f-1_Af\rvert>\varepsilon\right\}\\\subset \left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_{A_n}f\rvert+\lvert 1_{A_n}f-1_Af\rvert>\varepsilon\right\}\\\subset\left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_{A_n}f\rvert>\frac{\varepsilon}{2}\right\}\cup\left\{\omega\in\Omega:\lvert 1_{A_n}f-1_Af\rvert>\frac{\varepsilon}{2}\right\}\\=\left\{\omega\in\Omega:\lvert 1_{A_n}(f_n-f)\rvert>\frac{\varepsilon}{2}\right\}\cup\left\{\omega\in\Omega:\lvert f(1_{A_n}-1_A)\rvert>\frac{\varepsilon}{2}\right\}\\\subset\left\{\omega\in\Omega:\lvert f_n-f\rvert>\frac{\varepsilon}{2}\right\}\cup\underbrace{\left\{\omega\in\Omega:\lvert 1_{A_n}-1_A\rvert=1\right\}}_{=\left\{\omega\in\Omega:\lvert 1_{A_n}-1_A\rvert>\lambda\right\}\text{ for a }0<\lambda<1}.
$$
So it is
$$
\mu(\left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_Af\rvert>\varepsilon\right\})\leq\mu(\left\{\omega\in\Omega:\lvert f_n-f\rvert>\frac{\varepsilon}{2}\right\}\cup\left\{\omega\in\Omega:\lvert 1_{A_n}-1_A\rvert>\lambda\right\})\\\leq\underbrace{\mu(\left\{\omega\in\Omega:\lvert f_n-f\rvert>\frac{\varepsilon}{2}\right\})}_{\to 0}+\underbrace{\mu(\left\{\omega\in\Omega:\lvert 1_{A_n}-1_A\rvert>\lambda\right\})}_{\to 0}\to 0
$$
$$
\Longrightarrow \mu(\left\{\omega\in\Omega:\lvert 1_{A_n}f_n-1_Af\rvert>\varepsilon\right\})\to 0
$$
From I. and II. it follows that $(1_{A_n}f_n)_{n\in\mathbb{N}}$ converges to $1_Af$ in mean.
Could you pls say me if my proof is allright?
With best wishes for X-mas,
math12
No comments:
Post a Comment