I was wondering what does the multiplication of two Lebesgue integrable simple functions look like. Assume integral of a function f is defined as

\sum _{{k=1}}^{n}a_{k}\mu (A_{k}),

  • $\begingroup$ The product of two measurable simple functions is a measurable simple function. $\endgroup$ – MPW May 23 at 11:26

If $\mathbb{1}_A$ stands for the indicatrice function of the measurable set $A$, that is takes values $1$ and $0$ and is $1$ exactly on $A$, then

\begin{align} \mathbb{1}_A \times \mathbb{1}_B = \mathbb{1}_{A\cap B} \end{align}

Use that to check the result on simple functions: the product of two simple functions is then a simple function.

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