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I am writing a document with some mathematics and I need help with the construction of a sentence.
Is it correct to write "the function f measures the failure of the space M to be smooth"? I am not sure if there is a mistake concerning the verb "to be", it does not sound natural to me.
... failure of the space M to be smooth is indeed grammatical English, but I'm whether it's the usual way to express what you want is a question for Mathematics SE.
Usually you see an abstract quality being measured, such as "smoothness" or "roughness":
Here are a couple of examples from published papers:
Smoothness:
... certain geometric measures of smoothness ... (https://www.jstor.org/stable/3212668)
Roughness::
It is well-known that random functions are typically “rough” (non-differentiable), which raise the question of determining and measuring roughness. Probably, the most studied roughness measures are the Hausdorff dimension and the p−variation index. (https://arxiv.org/pdf/0802.0489.pdf)
Lack-of-smoothness:
Multidimensional Whittaker graduations are obtained by minimizing a linear combination of a measure of lack-of-fit between the graduated and ungraduated values, and measures of lack-of-smoothness of the graduated values across each dimension. (https://www.sciencedirect.com/science/article/pii/0167668795000216)
