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Show that the product of two step functions is a step function

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Show that the product of two step functions is a step function.

Notes from section of book attached.

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In this solution, it is shown that the product of two step functions is a step function. The solution is detailed and well presented.

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observe first that for subsets A, B of |R (or some ambient set X)

(chi_A)(chi_B) = chi_(A x B)

where chi_A denotes the characteristic function of A, etc.

Now, suppose f, g are step functions on subsets A, B of the real line, respectively; then they have the ...

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