Answer to Question #235644 in Real Analysis for Kevin

Question

Measure theory

1)find f+ and f- if fx =cosx+1/2 0<x<2π

2)find the measure of the set{x:sinx>=1/2} for 0<x<2π

Answer 1

1) Consider the inequality $\cos x+1/2>0$ for $0<x\leq \pi$. On this interval cosine is a monotonely decreasing function, $\cos x>-1/2=\cos(2\pi/3)$ iff $0<x<2\pi/3$.

Since $\cos(2\pi-x)=\cos x$, for $x\in(\pi,2\pi)$ $\cos x>-1/2$ iff $0<2\pi-x<2\pi/3$, or $4\pi/3<x<2\pi$.

Finally, for $x\in(0,2\pi)$,

if $x\notin(2\pi/3,4\pi/3)$ then $(\cos x+1/2)_+ =\cos x+1/2$ and $(\cos x+1/2)_- =0$.

if $x\in(2\pi/3,4\pi/3)$ then $(\cos x+1/2)_- =\cos x+1/2$ and $(\cos x+1/2)_+ =0$.

2) $\{x\in(0,2\pi):\sin x\geq 1/2\}=(\pi/6, 5\pi/6)$, therefore,

$\mu(\{x\in(0,2\pi):\sin x\geq 1/2\})=\mu((\frac{\pi}{6}, \frac{5\pi}{6}))=\frac{5\pi}{6}-\frac{\pi}{6}=\frac{2\pi}{3}$