Hi

I have been working with the topic Lebesgue integration for some time, but I have some questions.

Lets say the the function $\displaystyle f(x) = sin^{-\beta}(x)$ where $\displaystyle \beta \in \mathbb{R}$ and $\displaystyle x \in ]0, \frac{\pi}{2}[$

I will like to show for which values of $\displaystyle \beta$ that this function f(x) is integratable in 0 and $\displaystyle \frac{\pi}{2}$ and finally can be integrated over the $\displaystyle ]0, \frac{\pi}{2}[$

I have a feeling that first I first need to show that f(x) is a messurable function which is also know as a Borel function. I know that my function is mesurable if its continous. Since the preimage of f(x) is open, then this is obviously continous, and therefore a Borel function.

According to the definion I know of Local Lebesgue integratable functions, then if

$\displaystyle f: B \rightarrow \mathbb{C}$ is called Lebesgue Integratable if $\displaystyle f|B \in \mathcal{L}(K)$ for every compact subset subset $\displaystyle K \subseteq B$

Thus $\displaystyle \int_{K} |f(x)| dx < \infty$for every compact subset $\displaystyle K \subseteq B$

Could someone here please give me a hint how I by applying this definition Lebesgue integratable function on my function can find the desired values of $\displaystyle \beta$??

best Regards

Billy