= Lebesgue integration
{wiki}
<Lebesgue integration> builds an <integral> from <measures> of <measurable sets>. On a <measure space> $(X,\mathcal A,\mu)$, a nonnegative <simple function> $s=\sum_{j=1}^m a_j\mathbf1_{E_j}$ on disjoint <measurable sets> has
$$
\int_Xs\,d\mu=\sum_{j=1}^m a_j\mu(E_j),\qquad a_j\ge0.
$$
Here $\mathbf1_{E_j}$ is the <indicator function> of $E_j$, and $0\cdot\infty$ is taken as zero. For any nonnegative <measurable function> $f$, define its <Lebesgue integral> by approximation from below:
$$
\int_Xf\,d\mu=\sup\left\{\int_Xs\,d\mu: s\text{ is a nonnegative simple function},\ s\le f\right\}.
$$
This value may be infinite. For a signed <measurable function>, write $f=f_+-f_-$ using its <positive part> and <negative part>. A finite <Lebesgue integral> exists exactly when $\int_X|f|\,d\mu<\infty$; subtract the two finite contributions. One must not subtract two infinite values.<Measure Theory by John K. Hunter>[Chapter 4, Definitions 4.1, 4.4 and 4.8]{ref}
For an explicit <simple function> on $[0,1]$, take $f=2$ on $[0,1/3)$ and $f=5$ on $[1/3,1]$. Its <Lebesgue integral> is $2(1/3)+5(2/3)=4$: values are weighted by the <measures> of the sets carrying them. A set of <Lebesgue measure> zero contributes zero, so changing a <function> on a <null set> does not change its <Lebesgue integral>.
The <monotone convergence theorem> and <dominated convergence theorem> make <Lebesgue integration> useful when passing from approximations to a <limit>. For the latter, <measurable functions> $f_n$ converging <almost everywhere> to $f$ satisfy $\int f_n\,d\mu\to\int f\,d\mu$ when $|f_n|\le g$ for a fixed <integrable function> $g$. The common bound prevents a shrinking peak from retaining a nonzero <integral> while disappearing pointwise.
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