Littlewood's three principles

Remarks

The precise forms on a set $E \subseteq \R$ with $\lambda(E) < \infty$ are as follows. For principle 1: for every $\varepsilon > 0$ there is a finite union $U$ of intervals with $\lambda(E \mathbin{\triangle} U) < \varepsilon$. For principle 2: this is Lusin's theorem, which gives a closed $F \subseteq E$ with $\lambda(E \setminus F) < \varepsilon$ on which $f$ is continuous. For principle 3: this is Egorov's theorem, which gives uniform convergence outside a set of measure less than $\varepsilon$. Example: for $f = \mathbf{1}_{\Q}$ on $[0, 1]$, take $F$ a closed set of irrationals with $\lambda(F) > 1 - \varepsilon$; then $f|_F = 0$ is continuous. Without finite measure, principle 3 fails: on $\R$, the functions $\mathbf{1}_{[n, \infty)}$ tend to $0$ pointwise, but not uniformly outside any set of finite measure.