Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can find a norm $\|\cdot\|$ on the space of measurable functions (or at least a subspace of "sufficiently nice" functions) satisfying the following two conditions?


1. $\|f\|\le \|g\|$ if $|f|\le |g|$ (pointwise).
2. $\|\chi\_A\|=\phi(\mu(A))$, where $\chi\_A$ is the characteristic function.


Such a norm seems to be constructible by integrals if $\phi$ is simple. E.g. $\phi(u)=\sqrt{u+u^2}$, we can take $\|f\|=\sqrt{(\int\_X |f|d\mu)^2+\int\_X |f|^2 d\mu}$. But otherwise I am stuck. If this is not true in general, what conditions on $\phi$ can we ensure such an existence?



Existence of a certain norm on space of measurable functions
What you are asking about is known as a **symmetric** (or rearrangement invariant) Banach space of measurable functions (not to be confused with Riemannian symmetric spaces) and the associated **fundamental function**, see the recent book [Foundations of symmetric spaces of measurable functions](https://mathscinet.ams.org/mathscinet-getitem?mr=3525091) by Rubshtein et al. For instance, the **Lorentz** Banach spaces are the ones that correspond to concave functions $\phi$ with $\phi(0)=0$. 


