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

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

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

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


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

  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

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

    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.

Richard B. Melrose 2004-05-24