It is a well-known fact that for any compact topological space $\rm S$, the set its points is in bijection with the set of maximal ideals of $\mathcal C(\rm S)$, the algebra of continuous functions $\rm S \to \mathbb R$.

Now consider the circle $\rm S^1$ with its usual $\sigma$-algebra. Note $\mathcal M$ the algebra of measurable functions $\rm S^1 \to \mathbb R$ and $\mathcal N$ the ideal of functions vanishing almost everywhere.

By Zorn's lemma, the algebra $\mathcal M / \mathcal N$ has at least one maximal ideal. However in this example, it is not possible to get a maximal ideal by considering functions vanishing at one point. 

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> What is an example of a maximal ideal of this algebra?
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