Exercise 1.5

Question

\begin{enumerate}[label=(\alph*)]
      \item Suppose $f\colon X 	o [-\infty,\infty]$ and $g\colon X 	o
        [-\infty,\infty]$ are measurable. Prove that the sets
        \begin{equation*}
          \{x : f(x) < g(x)\},\ \{x : f(x) = g(x)\}
        \end{equation*}
        are measurable.
      \item Prove that the set of points at which a sequence of measurable
        real-valued functions converges (to a finite limit) is measurable.
    \end{enumerate}

Amit Awasthi · Accepted Answer

\begin{proof}[Proof of $(a)$]
      The first set is $(g - f)^{-1}((0,\infty])$, and the second is
      $(g-f)^{-1}(0)$, both of which are measurable.
    \end{proof}
    \begin{proof}[Proof of $(b)$]
      This set can be written as
      \begin{equation*}
        \bigcap_{k=1}^\infty \bigcup_{N = 1}^\infty \bigcap_{m,n \ge N} \{x \in X
        \mid \lvert f_m(x) - f_n(x) \rvert < 1/k \}.\qedhere
      \end{equation*}
    \end{proof}

Exercise 1.5

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Exercise 1.5

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