Theorem 6.16 in Baby Rudin: Is this theorem also valid for vector valued functions?

Here is the link to my earlier post here on Math Stack Exchange on Theorem 6.16 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:

Theorem 6.16 in Baby Rudin: $\int_a^b f d \alpha = \sum_{n=1}^\infty c_n f\left(s_n\right)$

Now my question is, Is this theorem valid for vector-valued functions too?

My feeling is that the answer to this question is in the affirmative. Am I right?

Here is my Theorem 6.16 in Baby Rudin for vector-valued functions:

Suppose $\sum c_n$ is a convergent series of non-negative real numbers, $\left( s_n \right)$ is a sequence of distinct points in $[a, b]$, and $$\alpha(x) = \sum_{n=1}^\infty c_n I\left(x-s_n \right)$$ for all $x \in [a, b]$. Let $\mathbf{f}$ be a continuous mapping of $[a, b]$ into $\mathbb{R}^k$. Then $\mathbf{f} \in \mathscr{R}(\alpha)$ on $[a, b]$, and $$\int_a^b \mathbf{f} \ \mathrm{d} \alpha = \sum_{n=1}^\infty c_n \mathbf{f} \left( s_n \right).$$

And, here is my proof:

Let $\mathbf{f} \colon= \left( f_1, \ldots, f_k \right)$, where each $f_j$ is a real function defined on $[a, b]$. As $\mathbf{f}$ is continuous on $[a, b]$, so is each $f_j$; therefore each $f_j \in \mathscr{R}(\alpha)$ on $[a, b]$, which implies that $\mathbf{f} \in \mathscr{R}(\alpha)$ on $[a, b]$ also, and \begin{align} \int_a^b \mathbf{f} \ \mathrm{d} \alpha &= \left( \int_a^b f_1 \ \mathrm{d} \alpha, \ldots, \int_a^b f_k \ \mathrm{d} \alpha \right) \qquad \mbox{ [ by Definition 6.23 in Baby Rudin ] } \\ &= \left( \sum_{n=1}^\infty c_n f_1 \left( s_n \right) , \ldots , \sum_{n=1}^\infty c_n f_k \left( s_n \right) \right) \\ & \qquad \qquad \mbox{ [ by Theorem 6.16 in Baby Rudin applied to each f_j ] } \\ &= \end{align}

What next?

Can we go from here to our desired answer?

Or, is this result not valid for vector-valued functions?

• You have it. Your last vector in your proof is precisely $\sum c_n\mathbf f(s_n)$. – Ted Shifrin Jul 25 '17 at 20:08