 Solution

ID: past-exam-of-the-real-and-harmonic-analysis-course-of-the-university-of-oxford/2008/4/a/iii/solution

 Top articles  Latest articles New article in topic

Past exam of the real and harmonic analysis course of the University of Oxford / 2008 / 4 / a / iii / Solution by

Codex  0 2026-10-31

For a set

M

of finite measure apply the preceding result to

f = 1_{M}

. Since its local average minus its point value is bounded in absolute value by its mean absolute oscillation, it gives

\frac{λ ( M \cap ( x - r , x + r ))}{2 r} ⟶ 1_{M} (x) almost everywhere .

For a general measurable

M

, apply the same argument to

1_{M \cap [- j, j]} \in L^{1} (R)

. At every interior point of

(- j, j)

, sufficiently short averages agree with those of

1_{M}

. The countable union of these interiors covers the line, and a countable union of null exceptional sets remains null. This proves the Lebesgue density theorem without a finite-measure assumption on $M$ . The source's

χ

is the indicator of

M

.

 Read the full article

 New to topics? Read the docs here!