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Stupid Question: If I'm given a non-euclidean space, such as $S^n $ (n-dimensional unit sphere), with the the uniform measure $\mu$ on it.

I'm also given a function $f:S^n \to \mathbb{R} $ that is bounded, by a constant $C$ . As an analogous to the Lebesgue measure case, can I say that: $ \int_{S^n } f d\mu \leq C \mu ( S^n) = C $?

Hope you'll be able to help

Thanks !

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    $\begingroup$ Yes. This follows from the fact that the integral of a non-negative function is non-negative and is true for any measure on any measure space whatsoever. $\endgroup$ – Qiaochu Yuan Jul 22 '12 at 18:58
  • $\begingroup$ Great! THanks a lot ! $\endgroup$ – joshua Jul 22 '12 at 19:13
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Yes. This follows from the fact that the integral of a non-negative function is non-negative and is true for any measure on any measure space whatsoever.

Specifically, we have $C\ge f$ everywhere, so $C-f\ge 0$, and by the above observation

$$\int C-f d\mu \ge 0 \Rightarrow C\mu(S^n) = \int C\ d\mu \ge \int f d \mu.$$

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Another way to see this is that it follows directly from the definition of the integral of an arbitrary nonnegative measurable function: $\int f\,d\mu$ is the supremum of the integrals of simple functions that are bounded by $f$. Each of those simple functions is also bounded by $C$, so its integral is bounded by $C\mu(S^n)$. (If $f$ has a nontrivial negative part, removing it only raises the integral, so we can assume $f$ is nonnegative.)

Of course this assumes $\mu(S^n)$ is finite, but so does the question. If the measure is infinite then so is $\int C\,d\mu$ and the inequality is trivial.

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While the inequality $\int_{S^n}f d\mu \leq C \mu(S^n)$ seems correct, as several have explained, the equality $C \mu (S^n) = C$ seems mistaken. The question only mentions a uniform measure $\mu$ on the unit $n$-sphere, but says nothing else about $\mu$. So in theory, $\mu(S^n)$ might equal 1, in which case $C \mu (S^n) = C$, but it might also equal any other positive number, in which case the equality would fail. For example, it is common to scale $\mu$ so that $\mu(S^n) = \frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}$.

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