I need help with this real analysis question. First some introduction...
We say that $\Omega$ is a topological space and define $\mathcal{L}^\infty$ as the set of all bounded measurable functions $f: \Omega \rightarrow \mathbb{K}$ (where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$).
I am told to let $||f||_\infty$ refer to the "sup-norm" of $f \in \mathcal{L}^\infty$ (which I believe is just $||f||_\infty = sup\{|f(x)|: x \in \Omega\}$?)
$\mathcal{N}(\mu)$ is the set of all $\mu$-nullfunctions $g: \Omega \rightarrow \mathbb{K}$.
Let now $L^\infty(\mu) = \mathcal{L}^\infty + \mathcal{N}(\mu)$ and define $$||[f]||_{L^\infty(\mu)} = inf_{h \in [f] \cap \mathcal{L}^\infty}||h||_\infty $$
The question is how do I prove that for $[f] \in L^\infty$ there exists $g \in \mathcal{N}(\mu)$ such that the norm can be written as $$ ||[f]||_{L^\infty(\mu)} = || f + g||_\infty$$
? I have gotten the hint that I have to use " $inf\{t\geq 0 : f1_{\{|f|>t\} } \in \mathcal{N}(\mu) \}$ ", but I'm not sure how. I have tried to write $ M = inf\{t\geq 0 : f1_{\{|f|>t\} } \in \mathcal{N}(\mu) \}$ and then say $f = f1_{\{|f| > M\}} + f1_{\{|f| \leq M \} }$ so that I can define $g = f1_{\{|f| > M\}}$ as the nullfunction in question... but I don't know how to progress and I don't understand what is meant by the second $f$ in $ ||[f]||_{L^\infty(\mu)} = || f + g||_\infty$. Is it just some element within $[f]$ or is it presupposed that we've picked an $f \in \mathcal{L}^\infty$ and then let $[f]$ be the class of all functions that are equal almost everywhere to it?
Any help is appreciated