Riesz representation theorem for Bochner integral

Question

Suppose $X$ and $W$ are Hilbert space. Let $F:X\to W$ be a vector-valued function which is Bochner integrable. Define functional $L:W\to\mathbb{C}$ as follows $$L(f):=\int_X\langle f,F(x)\rangle_W d\mu(x).$$ Suppose $L$ is continuous linear functional on $W$. Then through Riesz representation theorem, there exists $g\in W$ such that $$\int_X\langle f,F(x)\rangle_W d\mu=\langle f,g\rangle_W$$ My question is under what assumption, $g$ will be equal to $\int_X F(x)d\mu(x)$, that is, $$\int_X\langle f,F(x)\rangle_W d\mu=\langle f,\int_X F(x)d\mu(x)\rangle_W$$ Any discussion is welcome! If it is possible, please recommend me some reference. Thanks in advance!

Answer 1

For Bochner integrable $F$, there exists a sequence $(s_{n})$ of simple functions such that $\displaystyle\int s_{n}\rightarrow\int F$ in $W$ and $\displaystyle\int\|s_{n}-F\|\rightarrow 0$.

Suppose that $s_{n}=\chi_{E}w$, then it is easy to see $\displaystyle\int\left<f,s_{n}\right>_{W}=\left<f,\displaystyle\int s_{n}d\mu\right>_{W}$, and the general case follows by linearity.

Now $\displaystyle\int s_{n}\rightarrow\int F$ in $W$ implies that $\left<f,\displaystyle\int s_{n}d\mu\right>_{W}\rightarrow\left<f,\displaystyle\int F\right>$.

While $\displaystyle\int\|s_{n}-F\|\rightarrow 0$ together with Cauchy inequality gives $\displaystyle\int\left<f,s_{n}\right>\rightarrow\int\left<f,F\right>$.

So the equality $\displaystyle\int\left<f,F\right>=\left<f,\int F\right>$ holds.