research36 min studyLesson 835 in Mathematics

Rectifiable sets and approximate tangent planes

Separate almost-everywhere planar structure from singular crossings and measure multiplicity.

Before you begin

On this page

The question for today

What you’ll understand

State countable rectifiability with its null-set convention and local measure hypothesis.
Interpret approximate tangent planes through concentration in shrinking cones.
Distinguish a crossing density from density created by multiplicity on a single sheet.

A geometric set need not be one smooth surface to have useful tangents. Countable rectifiability asks whether almost all of it is covered by Lipschitz images of the correct dimension. This is an almost-everywhere condition, so isolated crossings and other negligible exceptional points need not destroy it. Densities and approximate tangent planes reveal how the set looks after zooming in, while multiplicity belongs to a measure placed on the set rather than to its point-set geometry alone.

Countably many parametrizations cover almost all the geometry

A Hausdorff-measurable set E is countably m-rectifiable if, outside an $\mathcal H^m$ -null subset, it lies in a countable union of Lipschitz images of subsets of m-space. Equivalent formulations use suitable whole-space extensions or countably many continuously differentiable patches up to null sets. For the density and tangent conclusions below, assume E has locally finite m-dimensional measure. Rectifiability alone does not assert that the set is closed, connected, embedded, or smooth at every point.

The null-set qualification is geometric, not merely Lebesgue-null in the ambient dimension. A curve in the plane is already planar-Lebesgue-null, so ignoring all such ambient null sets would erase the object we want to study. At the opposite extreme, a purely unrectifiable set meets every m-rectifiable set in zero m-measure. These notions distinguish structural properties at a specified dimension rather than describing how jagged a plotted sample appears.

\mathcal H^m\!\left(E\setminus\bigcup_{j=1}^{\infty}f_j(A_j)\right)=0,\qquad f_j:A_j\subseteq\mathbb R^m\to\mathbb R^n\text{ Lipschitz}.

The negligible remainder is measured with the same Hausdorff dimension as the parametrizing domains.↔ Scroll wide equations horizontally.

Approximate tangents allow a negligible amount of off-plane mass

An approximate tangent m-plane at x captures nearly all the local m-measure in every sufficiently narrow cone about that plane. More precisely, after dividing by the natural $r^m$ scale, the measure outside each fixed cone tends to zero. A positive local density condition prevents an empty or negligible set from satisfying the concentration requirement trivially. This is weaker than requiring every nearby point to lie close to a smooth graph, but strong enough for tangential calculus almost everywhere.

For a countably m-rectifiable set with locally finite m-measure, the normalized unweighted density equals one and the approximate tangent plane exists at almost every point of the set. These are one-way conclusions under stated assumptions, not a proof that any set with a vaguely finite density is rectifiable. Exceptional points may have several persistent branches or no unique approximate tangent. A finite union of smooth curves already supplies such an exception at a crossing.

\lim_{r\downarrow0}\frac{\mathcal H^m(E\cap B_r(x))}{\omega_mr^m}=1\quad\text{for }\mathcal H^m\text{-almost every }x\in E.

This density-one statement concerns the unweighted measure of a locally finite rectifiable set.↔ Scroll wide equations horizontally.

A measure can add multiplicity without changing the set

If a positive locally integrable multiplicity theta weights $\mathcal H^m$ on a rectifiable set, the density of the weighted measure is theta almost everywhere at the appropriate differentiation points. The underlying tangent plane remains the same. A density of two can therefore mean a double-weighted single sheet, but at an exceptional crossing it can also reflect two different unweighted directions. Density alone does not resolve that distinction at every point.

Tangents to a smooth graph are recovered from derivatives, and their metric stretch agrees with the area Jacobian. This connects rectifiability to the previous area formula rather than introducing a separate coordinate rule. When a current is later built on such a set, its orientation and integer multiplicity provide additional data. Rectifiability supplies a geometric carrier; it does not select those data or guarantee a small boundary.

\mu=\theta\,\mathcal H^m\!\llcorner E\quad\Longrightarrow\quad \Theta^m(\mu,x)=\theta(x)\quad\text{for }\mathcal H^m\text{-almost every }x\in E.

The weighted density statement uses positive locally integrable multiplicity and differentiation on the rectifiable carrier.↔ Scroll wide equations horizontally.

Work it through

Two crossing lines are rectifiable but not tangent at their crossing

Let E be the union of the horizontal line and the line $y=x$ in the plane. Compute its one-dimensional density at the origin and at nonzero points. Compare it with a horizontal line carrying constant multiplicity two.

Each line meets the radius-r ball at the origin in a diameter of length $2r$ . Their intersection is one point, of zero one-dimensional measure. The union therefore has local measure $4r$ , giving normalized density two.
At a nonzero point of either line, sufficiently small balls miss the other line, so the density is one and the ordinary line is its approximate tangent. At the origin no single line captures both persistent branches in every narrow cone, so there is no approximate tangent line there.
A horizontal line weighted by two also has density two at the origin, but it does have a unique horizontal tangent. Thus the two measures share a density value without sharing tangent structure. The crossing is a one-dimensional-null exception and does not contradict countable rectifiability.

\Theta^1(\mathcal H^1\!\llcorner E,0)=\lim_{r\downarrow0}\frac{4r}{2r}=2.

The exceptional point contributes two full branches, while the almost-everywhere density theorem concerns regular points.↔ Scroll wide equations horizontally.

The result

The crossing density is two, regular-point density is one, and a double-weighted single line shows why multiplicity and multiple directions must be separated.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Recover a graph tangent and its local stretch

Pause and Reset, then set coefficient a to 1 and position x to 1. The graph y=x² passes through (1,1) with slope two. Predict tangent direction (1,2) and length stretch √5 for a unit parameter increment in the linearized graph. The plotted tangent checks the slope, while its normalized Hausdorff density must be analyzed in geometric balls separately. This instrument is a smooth quadratic graph, not a rectifiability test for arbitrary sets. Reset and set both controls before a fresh comparison; edits preserve time.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA parameter-speed factor and a geometric density ratio are different quantities, even for the same smooth curve.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The union is countably one-rectifiable and locally finite. Its sole crossing is an exceptional H¹-null point; almost every other point has one line as tangent and density one.

Your turn

Try it yourself

For the graph $y=x^2$ at $(1,1)$ , give a unit tangent direction and the normalized density of its unweighted length measure. What is the density if that measure is multiplied by three?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Normalize the derivative vector (1,2). The graph’s speed does not replace the geometric-ball density denominator.

Compare with the solution

A unit tangent is $(1,2)/\sqrt5$ . The unweighted smooth-curve length measure has normalized one-dimensional density one. Multiplying the measure by three changes that density to three without changing the tangent direction. The factor $\sqrt5$ is the parameter-speed Jacobian, not the geometric density at the smooth point.

Keep the idea

Take it with you

Rectifiability covers almost all of a set in its own Hausdorff dimension.
Locally finite rectifiable sets have density one and approximate tangents almost everywhere.
Multiplicity changes measure density without necessarily changing the tangent geometry.

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