Problem 1428 (difficulty: 5/10)

Let \(\displaystyle \mathcal{T}\) be the collection of the sets \(\displaystyle [a,b)\times[c,d)\).

(a) Show that \(\displaystyle \mathcal{T}\) is a semi-ring.

(b) What ring does \(\displaystyle \mathcal{T}\) generate?

(c) Show that \(\displaystyle f:\mathcal{T}\to\R\) is additive if and only if there is \(\displaystyle g:\R^2\to\R\) for which \(\displaystyle f\big([a,b)\times[c,d)\big)=g(b,d)-g(a,d)-g(b,c)+g(a,b)\).

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