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The right triangle ${(x, y) \in [0, 1]^{2} : x + y \leq 1}$ has Jordan content $1 / 2$

Example

The triangle $T = {(x, y) \in [0, 1]^{2} : x + y \leq 1}$ is Jordan measurable and has content $1 / 2$ .

Facts & Assumptions

Given: Uniform $N$ -by- $N$ grids with $N \geq 1$ .

[L1]

Jordan content equals the indicator integral (A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content).

[L2]

Each of the three edges is a continuous graph, after exchanging coordinates for the vertical edge, and their finite union has content zero (The graph of a continuous function on a closed nondegenerate rectangle in $R^{m}$ has content zero in $R^{m + 1}$ , A bounded set in $R^{m}$ is Jordan measurable iff its boundary is null, equivalently of content zero).

Verification

technique · induction

1.1

Index the grid cells by $0 \leq i, j < N$ . A cell is contained in $T$ when $i + j \leq N - 2$ , while it meets $T$ when $i + j \leq N$ . Thus the lower staircase has $N (N - 1) / 2$ cells and the upper staircase has $(N^{2} + 3 N - 2) / 2$ cells.

given

1.2

Induction gives $\sum_{k < N} ι (k) = ι (N) ι (N - 1) / 2$ . Hence the lower area is $(N - 1) / (2 N)$ , the upper area is $1 / 2 + 3 / (2 N) - 1 / N^{2}$ , and their gap is $2 / N - 1 / N^{2} < 2 / N$ . Both areas tend to $1 / 2$ .

baseihgiven

2.1

Steps 1.1 and 1.2 give inscribed and covering grid approximations converging to $1 / 2$ , so the inner and outer contents agree at $1 / 2$ . Alternatively, [L2] gives Jordan measurability from the boundary criterion. In either route, [L1] identifies the common value with the indicator integral.

step 1.1step 1.2L1L2discharge-induction∎

Depends on

Used by

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52 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)
A. Cañez, multivariable calculus notes (standard reference, not scraped)

The right triangle {(x,y)∈[0,1]2:x+y≤1}{(x,y)∈[0,1]2:x+y≤1} has Jordan content 1/21/2