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## Measure of open intervals and half-open half-closed intervals

Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A6 Prove that if $a,b\in\mathbf R$ and $a<b$, then $$|(a,b)|=|(a,b]|=|[a,b)|=b-a.$$ Solution 1: Since $(a,b)\subset [a,b]$, by 2.5 and…

## Measure on difference of two sets

Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A3 Prove that if $A,B\subset\mathbf R$ and $|A|<\infty$, then $$|B\setminus A|\geqslant |B|-|A|.$$ Solution: Note that we have B\subset…

## Union with a zero measure set does not alter the measure

Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A1 Prove that if $A$ and $B$ are subsets of $\mathbf R$ and $|B|=0$, then $|A\cup B|=|A|$. Solution:…