A step has no locally integrable weak derivative

Given: Countable Choice, $I=(-1,1)$, $E=(0,1)$, the indicator $H={\mathbf{1}}_E$, and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$.

Countable Choice, or ${\mathrm{A}\mathrm{C}}_{\omega }$, says that every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

The indicator of a measurable set is measurable. (An indicator function is measurable exactly when its set is measurable)

Under Countable Choice, intervals in $\mathbb{R}$ are measurable with their length as measure, including open and closed endpoint conventions; degenerate intervals have measure zero. (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 compact subset of $\mathbb{R}$ is measurable and has finite Lebesgue measure. (Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure)

The complex $L^p$ conventions use $\int \mathrm{\mid }f{\mathrm{\mid }}^p$ for finite $p$ and the essential bound for $p=\mathrm{\infty }$. (Complex Lp classes and Euclidean test-function conventions)

The simple integral of ${\mathbf{1}}_E$ is $\lambda (E)$, and the nonnegative Lebesgue integral agrees with that simple integral. (The integral of a nonnegative simple function, The nonnegative integral agrees with the simple integral on simple functions)

If $f\le g$ are nonnegative measurable functions, then $\int f\le \int g$. (Monotonicity and nonnegative homogeneity of the nonnegative integral)

For a measurable set $A$, the integral over $A$ is the integral of the integrand multiplied by ${\mathbf{1}}_A$. (Integral over a measurable subset)

A complex measurable function is integrable when its modulus is integrable, and its integral is defined componentwise. (Integrable real and complex functions, and their integrals)

A test function in $\mathcal{D}(I)=C_c^{\mathrm{\infty }}(I;\mathbb{C})$ is smooth and has compact support in $I$; its zero extension is smooth on $\mathbb{R}$. (Test function space d of an open set)

The regular distribution of a locally integrable function $u$ is $T_u(\phi )={\int }_{I}u\phi$. (Locally integrable functions as regular distributions)

The distributional derivative obeys $⟨\mathrm{\partial }T,\phi ⟩=-⟨T,{\phi }^{\mathrm{\prime }}⟩$. (Distributional derivative)

The Dirac distribution is ${\delta }_0(\phi )=\phi (0)$. (Dirac delta and its derivatives)

A locally integrable weak derivative $v$ satisfies ${\int }_{I}H{\phi }^{\mathrm{\prime }}=-{\int }_{I}v\phi$ for every $\phi \in C_c^{\mathrm{\infty }}(I)$. (Weak derivative of a locally integrable function)

Membership in $W^{1,p}$ requires an $L^p$ class with a locally integrable representative satisfying the first-order weak test identity. (Integer-order Sobolev spaces and their norms)

If $f\in L^1(\mu )$, then for every $\epsilon >0$ there is $\delta >0$ such that $\mu (A)<\delta$ implies ${\int }_A\mathrm{\mid }f\mathrm{\mid }d\mu <\epsilon$. (Absolute continuity of the integral)

There is a smooth $\eta :\mathbb{R}\to [0,1]$ equal to $1$ on $[-1\mathrm{/}2,1\mathrm{/}2]$ with support contained in $(-1,1)$. (A smooth bump between concentric Euclidean balls)

Composing smooth functions with the affine map $x\mapsto x\mathrm{/}ϵ$ preserves smoothness by the chain rule. (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$)

For complex $C^1$ functions on $[a,b]$, Countable Choice gives the Lebesgue fundamental theorem ${\int }_a^b{u}^{\mathrm{\prime }}=u(b)-u(a)$. (Complex integration by parts on intervals and decaying lines)

For an integrable complex function $f$, $\mid \int f\mid \le \int \mathrm{\mid }f\mathrm{\mid }$. (The modulus of an integral is bounded by the integral of the modulus)

A nonnegative integral over a measurable null set is zero. (A nonnegative integral over a null set vanishes)

Local integrability means finite integral of $\mathrm{\mid }f\mathrm{\mid }$ on each compact set. (Complex Lp classes and Euclidean test-function conventions)