I was interested in the concept of direct integrals. The definition, i was concerned with, can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of separable Hilbert spaces $(H(\omega))_{\omega \in \Omega}$ and a $\sigma$-finite measure space $(\Omega, \mu)$. The author uses sections $s: \Omega \to \cup_{\omega\in\Omega} H(\omega) $ with $s(\omega) \in H(\omega)$ and constructs (if the dimension map is measureable) a measureable set of (pointwise) orthonormal sections, in the sense that the scalar products with these are measureable. Then, the set of square integrable, measureable sections (modulo null sets) define the direct integral.

Later on, the same author uses the fact, that direct integrals and direct sums interchange: $$\int_{\Omega}^{\oplus} \bigoplus_{i=1}^\infty H^i_\omega \; d \mu (\omega) = \bigoplus_{i=1}^\infty \int_{\Omega}^{\oplus} H^i_\omega \; d \mu (\omega).$$

Is there a way to see that easily? Or did i get it wrong and the statement is generally not true?

Thank you and Greetings. :)

I am interested in the concept of direct integrals. The definition I am concerned with can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of separable Hilbert spaces $(H(\omega))_{\omega \in \Omega}$ and a $\sigma$-finite measure space $(\Omega, \mu)$. The author uses sections $s: \Omega \to \cup_{\omega\in\Omega} H(\omega) $ with $s(\omega) \in H(\omega)$ and constructs (if the dimension map is measureable) a measureable set of (pointwise) orthonormal sections, in the sense that the scalar products with these are measureable. Then, the set of square integrable, measureable sections (modulo null sets) define the direct integral.

Later on, the same author uses the fact that direct integrals and direct sums interchange: $$\int_{\Omega}^{\oplus} \bigoplus_{i=1}^\infty H^i_\omega \; d \mu (\omega) = \bigoplus_{i=1}^\infty \int_{\Omega}^{\oplus} H^i_\omega \; d \mu (\omega).$$

Is there a way to see that easily? Or did I get it wrong and the statement is generally not true?

I was interested in the concept of direct integrals. The definition, i was concerned with, can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of Hilbert spaces $(H(\omega))_{\omega \in \Omega}$. The author uses sections $s: \Omega \to \cup_{\omega\in\Omega} H(\omega) $ with $s(\omega) \in H(\omega)$ and constructs (if the dimension map is measureable) a measureable set of (pointwise) orthonormal sections, in the sense that the scalar products with these are measureable. Then, the set of square integrable, measureable sections (modulo null sets) define the direct integral.

Later on, the same author uses the fact, that direct integrals and direct sums interchange: $$\int_{\Omega}^{\oplus} \bigoplus_{i=1}^\infty H^i_\omega \; d \mu (\omega) = \bigoplus_{i=1}^\infty \int_{\Omega}^{\oplus} H^i_\omega \; d \mu (\omega).$$

Is there a way to see that easily? Or did i get it wrong and the statement is generally not true?

Thank you and Greetings. :)

I was interested in the concept of direct integrals. The definition, i was concerned with, can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of separable Hilbert spaces $(H(\omega))_{\omega \in \Omega}$ and a $\sigma$-finite measure space $(\Omega, \mu)$. The author uses sections $s: \Omega \to \cup_{\omega\in\Omega} H(\omega) $ with $s(\omega) \in H(\omega)$ and constructs (if the dimension map is measureable) a measureable set of (pointwise) orthonormal sections, in the sense that the scalar products with these are measureable. Then, the set of square integrable, measureable sections (modulo null sets) define the direct integral.

Later on, the same author uses the fact, that direct integrals and direct sums interchange: $$\int_{\Omega}^{\oplus} \bigoplus_{i=1}^\infty H^i_\omega \; d \mu (\omega) = \bigoplus_{i=1}^\infty \int_{\Omega}^{\oplus} H^i_\omega \; d \mu (\omega).$$

Is there a way to see that easily? Or did i get it wrong and the statement is generally not true?

Thank you and Greetings. :)