There's a stronger version of that theorem due to Lyapunov. It is stronger because it concerns vectors of measures, and not only a single measure. It states that given a non-atomic vector measure (a collection of $n$ measures $\mu_1,\ldots, \mu_n$ where each measure is non-atomic) always has an image which is convex (in $\mathbb{R}^n$).

Unfortunately, I could never find a translation of his paper, so I can only link the version in russian. The main statements can be found in French at the end of the paper. There's also a paper of Halmos that proves the result.

I think the result for vector measures is much harder to prove, but if what you want is a reference so you don't have to prove the result yourself, it serves the purpose.

There's a stronger version of that (basic) theorem due to Lyapunov. It is stronger because it concerns vectors of measures, and not only a single measure. It states that given a non-atomic vector measure (a collection of $n$ measures $\mu_1,\ldots, \mu_n$ where each measure is non-atomic) always has an image which is convex (in $\mathbb{R}^n$).

Unfortunately, I could never find a translation of his paper, so I can only link the version in russian. The main statements can be found in French at the end of the paper. There's also a paper of Halmos that proves the result.

Maybe looking at the proof method or subsequent papers you can find the chain statement that you seek.