Is the following function Lebesgue measurable? $$f(x):= \frac{\sin x}{1 + x^2} $$The problem confuses me a bit, since it doesn't state where it wants it to be Lebesgue integrable. I figure that since $f$ is continuous on any interval $[a,b]$, further it is bounded by $1$, then it is Riemann-integrable (since the points where it is discontinuous is a Lebesgue null set), and since it is Riemann-integrable, it's definitely Lebesgue-integrable.
Is this correct? If yes, how do I extend "Lebesgue integrable on $[a,b]$" to just "Lebesgue integrable"?