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 $t\to \operatorname{tr}(f(t))$ is Borel measurable.

Q. Suppose that $\int \operatorname{tr}(f(t))d\mu$ is finite. Can we conclude that $\int \operatorname{tr}(|f(t)|)d\mu$ is finite too?