Problem 1
Give an example of a set in \(R^{2}\) which is not a direct product of any two sets in \(R^{1}\).
Problem 3
Given three spaces \(X, \mathrm{Y}\) and \(Z\), equipped with measures \(\mu_{x}, \mu_{y}\) and \(\mu_{z}\), respectively, prove that \(\left(\mu_{x} \otimes \mu_{y}\right) \otimes \mu_{x}\) and \(\mu_{x} \otimes\left(\mu_{y} \otimes \mu_{z}\right)\) are identical measures on \(X \times Y \times Z\).
Problem 4
Let \(A=[-1,1] \times[-1,1]\) and $$ f(x, y)=\frac{x y}{\left(x^{2}+y^{2}\right)^{8}} $$ Prove that a) The iterated integrals (20) exist and are equal; b) The double integral (19) fails to exist. Hint. Since $$ \int_{-1}^{1} \mathrm{f}(x, y) d x=\int_{-1}^{1} \mathrm{f}(x, y) d y=0 $$ we have $$ \int_{-1}^{1}\left(\int_{-1}^{1} f(x, y) d x\right) d y=\int_{-1}^{1}\left(\int_{-1}^{1} f(x, y) d y\right) d x=0 $$ On the other hand, the double integral fails to exist, since $$ \int_{-}^{1} \int_{-1}^{1}|f(x, y)| d x d y>\int_{0}^{1} d r \int_{0}^{\imath \pi} \frac{|\sin 0 \cos \theta|}{r} d \theta=2 \int_{0}^{1} \frac{\mathrm{dr}}{\mathrm{r}}=\infty $$ after transforming to polar coordinates.
