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up vote 1 down vote favorite Are projective measurement bases always orthonormal? measurement share | improve this question edited 36 mins ago Blue ♦ 5,605 1 12 50 asked 58 mins ago ahelwer 1,143 1 12 Related: What is the difference between general measurement and projective mea

up vote 0 down vote favorite Suppose $f: mathbb R^n to mathbb R$ is a measurable function such that $|f(x)| leq g(x)$ , where $g(x) = c|x|^{-p} chi_{B(0,1)}(x)$ for some $c > 0, p <n$ . Prove $f$ is integrable. This is from Bass exercise 11.21. Let $tilde x=(x_1,...,x_{n-1}), x = (tilde x, x_n)$ , and $f^{x_n}(tilde x) = f(x) : mathbb R^{n-1} to mathbb R$ . If $p leq 0$ then $g$ is integrable, so that $f$ also is, and we are done. So assume $p > 0$ . We are suggested to proceed by induction on $n$ . The case for $n=1$ can be solved. Now assume the case holds for $n-1$ , and let $epsilon > 0$ be so small that $p+epsilon$ is still smaller than $n$ . Then $p-1 + epsilon < n-1$ , and we can show that $|f^{x_n}(tilde x)| leq c|tilde x|^{-p+1-epsilon} chi_{tilde B(0,1)}(tilde x) |x|^{-1+epsilon}$ . A per