Let $H$ be a separable Hilbert space and $L^{1}(H)$ be the space of trace class operators on $H$.
Let $f:\mathbb{R}\to L^{1}(H)$ be a measurable function that is $t\to tr(f(t))$ id$t\to \operatorname{tr}(f(t))$ is Borel measurable.
Q. Suppose that $\int tr(f(t))d\mu$$\int \operatorname{tr}(f(t))d\mu$ is finite. Can we conclude that $\int tr(|f(t)|)d\mu$$\int \operatorname{tr}(|f(t)|)d\mu$ is finite too?
