Measure Theory-Integration Over Non-Lebesgue Measures 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 ! 
 A: 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
$$\int C-f d\mu \ge 0 \Rightarrow C\mu(S^n) = \int C\ d\mu \ge \int f d \mu.$$
A: 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.
A: 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 $\mu(S^n) = \frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}$.
