Riemann and Lebesgue integrals agree for continuous data on [0,1]

Given: A continuous real function $g$ on the compact interval $[0,1]$.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous real function on a compact interval is bounded and Riemann integrable; its Riemann integral lies between every lower and upper Darboux sum.

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length and Lower bound, bounded below, bounded set: $[0,1]$ is closed and bounded. A subset of $\mathbb{R}$ is compact if and only if it is closed and bounded therefore makes it compact, so Heine-Cantor in $\mathbb{R}$: a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous gives uniform continuity of $g$.

Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$, and The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$: for a partition, the oscillation sum is the upper sum minus the lower sum, and an integrable function's integral lies between those sums; every tagged sum lies between them.

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: under Countable Choice, every interval cell of length $1\mathrm{/}N$ in the uniform partition has Lebesgue measure $1\mathrm{/}N$; a finite endpoint has measure zero.

The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions, and The Lebesgue integral is linear on $L^1(\mu )$: a finite real simple function is Lebesgue integrable and its integral is the finite sum of its values times the measures of its level sets; finite sums and differences of integrable functions integrate linearly.

Monotonicity and nonnegative homogeneity of the nonnegative integral and Integrable real and complex functions, and their integrals: if $u$ is Lebesgue measurable and $\mathrm{\mid }u\mathrm{\mid }\le \epsilon {\mathbf{1}}_{[0,1]}$, then $u$ is integrable and $\mathrm{\mid }\int ud\lambda \mathrm{\mid }\le \epsilon$.

The Borel sigma-algebra of a topological space and Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point: preimages of open subsets of $\mathbb{R}$ under $g$ are relatively open in $[0,1]$, so preimages of all Borel sets are relative Borel and $g$ is Borel measurable.

Closure properties of measurable functions used by the integral: sums and differences of measurable real-valued functions are measurable.

The Borel sigma-algebra of a topological space and Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length: relative closed intervals and singletons in $[0,1]$ are Borel; each half-open cell $[a,b)$ is $[a,b]\setminus \{b\}$ and hence is Borel.

Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable: under Countable Choice, every Borel subset of $\mathbb{R}$ is Lebesgue measurable. A relative Borel subset of the closed interval $[0,1]$ is Borel in $\mathbb{R}$, so the same inclusion holds on $[0,1]$.

$g$ is continuous on $[0,1]$.

Countable Choice is assumed for the Lebesgue interval-measure formula [F4] and Borel-to-Lebesgue inclusion [F10]; the finite-partition comparison below uses no further choice principle.