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Big Rudin defines inner product as $(x,y)=\sum_{i=1}^n x_i \bar{y}_i$ in Example 4.5(a), and as $(f,g)=\int_X f\bar{g} d\mu$ in Example 4.5(b).Then Rudin says the former is a special case of the latter. At last, Rudin asks "What is the measure in (a)?" to conclude Example 4.5(b). I am puzzled to answer this question. Any detailed clearness? Thanks in advance.

I try to understand the measure as $\mu(p_i)=a_i$ with $\sum_{i=1}^n a_i=1$ and $a_i \ge 0$ like the interpretation to Theorem 3.3 of Jensen's Inequality. Then $\int_X x\bar{y} d \mu = \sum_{i=1}^n x_i \bar{y}_i a_i$. But $a_i$ does not appear in the inner product of Example 4.5(a).

There is a similar quetions of "relationship between discrete and continuous time inner product" at relationship between discrete and continuous time inner product. But I seem not to get the answer for my puzzle.

beginerXavier
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Let $\nu$ be counting measure on the set $X=\{1,2\}$. If $x,y\in\Bbb R^2$ and they're regarded as functions defined on $X$ (so $x(j)=x_j$) then $$x_1y_1+x_2y_2=\int_Xxy\,d\nu.$$

David C. Ullrich
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  • Thank David for clarifying my puzzle. Based on this information, I find a useful question and answer on integration of counting measure at https://math.stackexchange.com/questions/764076/integration-with-respect-to-counting-measure. Rober Ash and Catherine Doleans-Dabe's Probability and Measure Theory (2nd) provides also related elaboration on page89-90. Hope it be helpful for self-learners like me. – beginerXavier Jul 07 '22 at 15:19