- We developed some quantitative notions in hard analysis by refining the corresponding ones in soft analysis.
- We applied hard analysis to prove the following theorems and lemmas detailedly.
- Lebesgue approximation theorem for a set
- Lebesgue approximation theorem for a function
- Quantitative Lebesgue differentiation theorem
- Lebesgue regularity lemma
- Strong Lebesgue regularity lemma
