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I've seen similar questions to my question, as has been suggested by some responses to the first version, but the answers (e.g. from Christopher L), don't help me understand exactly why the Hausdorff measure decreases as delta increases. In particular I'm not able to see why the infimum of the sum of the diameters of the cover could become smaller.

Please could someone spell it out for me.

Thank you.

Paul Bratch
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    See here: https://math.stackexchange.com/questions/1982919/why-is-h-deltad-monotone-decreasing-in-delta-in-the-definition-of-hausdor?rq=1 – Rigel Sep 03 '17 at 10:06
  • Thank you for pointing me at that question, but that is the very question that I'm finding doesn't answer the bit I'm still stuck on. I'm still not able to see why the infimum of sum of the diameters of the cover can become smaller as delta increases. Is there anything further you could add to help me overcome this blockage? Thanks. – Paul Bratch Sep 03 '17 at 22:35
  • Your question is not directly related to Hausdorff measures. Simply, you have only to use the fact that, if $A,B\subset\mathbb{R}$ and $A\subset B$, then $\inf A \geq \inf B$. – Rigel Sep 04 '17 at 06:11
  • Thank you Rigel. While I'm able to see that what you say is true, it seems to me that the opposite is being claimed in the answer to the referenced Hausdorff measure question, i.e. the claim in that answer is that inf A <= inf B. That is the bit that is confusing me. Can can you help me understand that bit (which I'd agree is not a Hausdorff measure issue). Thanks again. – Paul Bratch Sep 04 '17 at 07:06
  • An example may help. Take $A = [0,1]$, $B = [-3, 5]$. You have $A\subset B$, $\inf A = 0 > -3 = \inf B$. When you compute $\inf B$ you have more elements wrt the ones you consider in the computation of $\inf A$. – Rigel Sep 04 '17 at 07:13
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    Finally it has dawned!!! My confusion was in thinking of smaller in absolute terms rather than more negative. Maybe that's because I've not done any maths for 35 years so some of the maths corners of my brain are a bit non-existent Thank you for your patience helping me over that hump. – Paul Bratch Sep 04 '17 at 10:53

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