Define $f_\alpha(x):=\frac{\Gamma(\frac{\alpha+1}{2})}{\sqrt{\alpha \pi} \Gamma(\frac{\alpha}{2})}(1+\frac{x^2}{\alpha})^{-\frac{\alpha+1}{2}}$ for $x\in \mathbb{R}$ and $\alpha >0$

Let $m$ be the Lebesgue measure. If $f_\alpha=f_\beta$ $m$-a.e., then how do I prove that $\alpha=\beta$?

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    Suppose that $\alpha \neq \beta$ and prove that $f_\alpha$ and $f_\beta$ differ on a set of Lebesgue measure > 0 – windircurse Dec 7 at 13:41
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    @windircurse Isn't that just a restatement of my question? How do I find such set? – Rubertos Dec 7 at 13:42
  • prove that the function $\alpha \to f_\alpha$ is decreasing – windircurse Dec 7 at 13:48
  • Could you be more specific? And the map $\alpha\to f_\alpha$ is not decreasing, as you can see here : en.m.wikipedia.org/wiki/Student%27s_t-distribution – Rubertos Dec 7 at 13:53

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