Practice for first test

I have decided to make the first test open-book, but only copies of Adams and Guillemin are permitted, no papers or other writings.

1. Set $X =[0,1]\times[0,1],$ the unit square in ${\mathbb{R}}^2$ and suppose that $f:X\longrightarrow [0,\infty)$ is Lebesgue measurable and such that

   $$\mu_{\text{Leb}}\{x\in X;f(x)\ge n\}\le 2^{-n}\ \forall\ n\in\mathbb{N}.$$

   
   
   Show that $\int_{X}fdx<\infty.$
2. Without using the equality of Riemann and Lebesgue integrals show that

   $$\int_{\bbR}(1+\vert x\vert)^{-1}dx=\infty$$

   
   
   and

   $$\int_{\bbR}(1+\vert x\vert^2)^{-1}dx<\infty.$$

   
   

3. Give, with proofs, an example of a non-negative Lesbesgue measurable function on $[0,1]$ which has finite Lebesgue integral but is not bounded.
4. Explain why the formula

   $$\mu(A)=\int_A\exp(-x^2)dx$$

   
   
   defines a countably additive finite measure on the $\sigma$-ring, $\mathcal M,$ of all Lebesgue measurable subsets of $\mathbb{R}.$ Show that completion of $\mathcal M$ with respect to this measure is just $\mathcal M$ again.
5. Let $(X,\mathcal F,\mu)$ be a measure space and suppose that $s_n:X\longrightarrow [0,\infty)$ is an increasing (i.e. pointwise non-decreasing) sequence of simple measurable functions with $\int_Xs_nd\mu<1.$ Show that the set consisting of those points $x\in X$ where $s_n(x)\to\infty$ has $\mu$-measure zero.