Approximation by simple functions: a confusion

lebesgue-measuremeasure-theoryreal-analysis

I am confused about the following matter (I come to a conclusion which is obviously wrong)

  1. if every real-valued function is a pointwise limit of a sequence of simple functions, e.g.
    \begin{equation*}
    f_n(x) = \left\{ \frac{j}{2^n} : j\in \mathbb{N}\cap [0,2^{2n}] \text { and } \frac{j}{2^n} \leq f(x) \right\}
    \end{equation*}
  2. and if simple functions are Lebesgue-measurable
  3. and if pointwise limit of Lebesgue-measurable functions is again Lebesgue-measurable

then does that mean that every real-valued function is Lebesgue-measurable?

Best Answer

The second statement is wrong. Consider the characteristic function of a set which is not Lebesgue measurable. The construction of a non-measurable set is classical. For instance, see https://www.math.purdue.edu/~zhang24/NonMeasurableSet.pdf

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