Given $f:(\mathbb{R},\mathbb{B}(\mathbb{R})) \rightarrow (\mathbb{R}, \mathbb{B}(\mathbb{R}))$ is a non-negative function, such that $\int_{\mathbb{R}}fd\mu < \infty$. Given $F:\mathbb{R} \rightarrow \mathbb{R}, F(x):= \int_{(-\infty, x)}f d\mu, \forall x \in \mathbb{R}$. Show that $F$ is continuous.
Notation used: $\mu$ denotes the lebsgue measure and $\mathbb{B}(\mathbb{R})$ denotes the Borel sigma-algebra.
What I've tried:
My initial thoughts were to attempt to use sequential continuity to prove this. We can define an increasing sequence of simple functions $(\phi_{n})_{n \in \mathbb{N}}$ such that $\lim_{n \to \infty} \phi_{n}(x) = f(x).$
My other thought was to try and use the Dominated Convergence Theorem (DCT), since we have a sequence of simple functions as above.
My issues with using DCT are:
- I'm not sure what to take as the dominating function;
- In DCT, it is the limit of the sequence of functions that we obtain as the result. But we already know what the sequence of simple functions limit is...
Thanks.
Your idea using DCT works. To show that $F$ is continuous in any $x_0\in\mathbb{R}$, consider a sequence $(x_n)$ with $x_n\rightarrow x_0$. Then the sequence of functions $h_n(t):=f(t)\,\chi_{(-\infty,x_n)}(t)$ converges pointwise a.e. to $h(t):=f(t)\,\chi_{(-\infty,x_0)}(t)$, and $|h_n(t)|\leq f(t)$. Since $f$ is integrable, DCT implies $$ \lim_{n\rightarrow\infty}F(x_n) = \lim_{n\rightarrow\infty}\int_{\mathbb{R}} h_n(t)\, dt = \int_{\mathbb{R}} h(t)\, dt = F(x_0). $$
Addition:
Verification that $h_n$ converges pointwise a.e. to $h$: Fix $t\in\mathbb{R}$. If $t>x_0$, then for $N$ sufficiently large we have $t>x_n$ for all $n>N$, and consequently $h_n(t)=0$ for all $n>N$. In this case we therefore have $\lim_{n\rightarrow \infty}h_n(t)=0=h(t)$. If $t<x_0$, then for $N$ sufficiently large $t<x_n$ for all $n>N$, and consequently $h_n(t)=f(t)$ for all $n>N$. In this case we therefore also have $\lim_{n\rightarrow \infty}h_n(t)=f(t)=h(t)$. We have thus established convergence of $h_n(t)$ to $h(t)$ for all $t\in\mathbb{R}\setminus\{x_0\}$. This means that $h_n$ converges pointwise to $h$ almost everywhere (the only exception being the null set $\{x_0\}$).