More detail please on exactly why the Hausdorff measure is monotone decreasing?
32117
Why is $H_\delta^d$ monotone decreasing in $\delta$ in the definition of Hausdorff measure?
40785
Finite signed measures and their boundedness
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
1794
$\exp(A+B)$ and Baker-Campbell-Hausdorff
2915
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
147546
$f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable
90540
What is wrong in this proof: That $\mathbb{R}$ has measure zero
132824
Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
147331
Compactness and Hausdorffness
19997
Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?
149390
Fixed point of a Function (monotonically increasing)
51223
Example of decreasing sequence of sets with first set having infinite measure
32347
Limit of a decreasing sequence of measurable sets.
102076
Sequence of monotone functions converging to a continuous limit, is the convergence uniform?
16743
Graph of real continuous function has measure zero
154240
Inverse of Heine–Cantor theorem