I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem:
Suppose that $f$ and $f_n$ are nonnegative measurable functions, that $f_n$ decreases pointwise to $f$, and that $\int_{\mathbb{R}}f_1 < \infty$. Prove that $\int_{\mathbb{R}}f = \lim\int_{\mathbb{R}}f_n$. [Hint: Consider $g_n = f_1 – f_n$]. Show with a counterexample that the assumption that $\int_{\mathbb{R}}f_1 < \infty$ is necessary.
The assumptions are (I always rewrite the problem to see if I understand it correctly):
- $\forall x \in \mathbb{R}, f_n(x) \geq f_{n+1}(x)$.
- $\int_{\mathbb{R}}f_1 < \infty$,that is $f_1 \in L^1(\mathbb{R})$.
- $f_n \to f$ pointwise in $\mathbb{R}$.
My work and thoughts:
As $f_n$ is monotone non-increasing, $g_n = f_1 – f_n$ is a monotone non-decreasing sequence of non-negative Lebesgue measurable functions, i.e $0 \leq g_1 \leq g_2 \leq \cdots \leq f_1 – f$ and $\lim g_n = f_1 – f$. Therefore, by the monotone convergence theorem
$$ \int_{\mathbb{R}}\lim_{n \to \infty} g_n = \int_{\mathbb{R}}(f_1 – f) = \lim_{n \to \infty}\int_{\mathbb{R}}(f_1 – f_n).$$
Because $\int_{\mathbb{R}}f_1 < \infty$ and $f_n \leq f_1$ for all $n \in \mathbb{N}$, this implies that $\int_{\mathbb{R}}f_n < \infty$ for all $n \in \mathbb{N}$.
This is where I'm stuck. I think I'm really close to the desired result. How do I continue from here? Also how do I show that the assumption that $\int_{\mathbb{R}}f_1 < \infty$ is necessary using a counterexample?
Best Answer
$$ \int_{\mathbb{R}}f_1 - \int_{\mathbb{R}}f =\int_{\mathbb{R}}(f_1 - f) =\int_{\mathbb{R}}\lim_{n \to \infty} g_n = \lim_{n \to \infty}\int_{\mathbb{R}}(f_1 - f_n) = \int_{\mathbb{R}}f_1 - \lim_{n \to \infty}\int_{\mathbb{R}}f_n$$
Subtract $\int_{\mathbb{R}}f_1$ from both sides and you are done.