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I start saying that I know this question has been already proposed a thousand times on this forum BUT I couldn't find a direct answer so I try asking on my own.

What's the intuitive meaning of this definition of measurable set?

$\cdot$ Let $X$ be a set and $\lambda^*$ an outer measure. Then $X$ is said to be measurable if for each $B\in X \lambda^*(B)=\lambda^*(B\cap X) + \lambda^*(B\cap X^C)$

Or equivalently if $\lambda_*(X)=\lambda^*(X)$.

I feel OK with the second definition because it looks kind of similar to the intuition in Riemann Jordan "measure", but why are the two equivalent?

Thanks.

Ubi.B
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Baffo rasta
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  • @Brahadeesh I remember I had already read that post but I did one of my own cause still it wasn't clear to me. – Baffo rasta Dec 12 '18 at 14:00
  • Oh, alright. It seemed to me that you haven't made your question more specific than the other one. Have you seen this too: https://math.stackexchange.com/q/2794210/279515 ? Perhaps it is more helpful? –  Dec 12 '18 at 14:03
  • At the end of the day I was able to understand what's the meaning of that definition so I think I'm fine already, but still thanks for your effort! – Baffo rasta Dec 12 '18 at 14:43
  • Glad to know :) No problem! –  Dec 12 '18 at 14:51
  • Perhaps you can write up your understanding in the form of an answer, too? That way it will benefit others who come across this question. –  Dec 12 '18 at 14:53
  • As soon as I have time I will do it – Baffo rasta Dec 13 '18 at 07:48
  • Thanks Baffo :) –  Dec 13 '18 at 07:49

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