Determine if $u(t) \in \mathcal{L}^1(\lambda)$

Question

I am looking at the measure space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$

I want to determine if the function u(t)=$\frac{sin(t)}{1+t^2} \in \mathcal{L}^1$

My ideas are/were:

1. $\frac{sin(t)}{1+t^2}\leq \frac{1}{1}=1$ and the using DCT (dominated convergence theorem). However I am not quite sure that the constant function $w(t)=1 \notin \mathcal{L}^1(\lambda)$ because the integral of the constant function is infinite on $\mathbb{R}$. Is this true?

2. Instead I wanted to dominate u(t) with a function w(t) $\in \mathcal{L}^1$. $\frac{sin(t)}{1+t^2}\leq \frac{1}{t^2}=w(t)$ . But the p-series only works for $t\geq1$ and we are looking at the entire real line. Also I am not sure if I can "convert" the p-series sum to a finite integral so that $w(t) \in \mathcal{L}^1$. Or maybe I need to show that u(t) and w(t) are both possible to approximate with simple functions

Would be appreciated to know if I am on the right track

Answer 1

You don't need DCT for this. Use the fact that $|\frac {\sin t} {1+t^{2}}| \leq \frac 1 {t^{2}}I_{|t|>1}+I_{|t|\leq 1}$.