Lyapunov convexity theorem

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Problem 3.5.1 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.5 "Some Applications"

Define measures μ1,μ2\mu_{1}, \mu_{2} on the unit circle by

dμ1=cos⁡θdθ,dμ2=sin⁡θdθ d \mu _{1} = \cos \theta d \theta , \quad d \mu _{2} = \sin \theta d \theta

and find the range of the measure μ=(μ1,μ2)\mu = \left(\mu_{1}, \mu_{2}\right).

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Problem 3.5.2 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.5 "Some Applications"

Construct two functions ff and gg on [0,1][0,1] with the following property: If

dμ1=f(x)dx,dμ2=g(x)dx,μ=(μ1,μ2) d \mu _{1} = f(x) d x, \quad d \mu _{2} = g(x) d x, \quad \mu = \left(\mu _{1}, \mu _{2}\right)

then the range of μ\mu is the square with vertices at (1,0),(0,1),(−1,0),(0,−1)(1,0),(0,1),(-1,0),(0,-1).

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References

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.