Union of two measure set is measurable II HINDI

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union of two measurable sets is again a measurable set. Theorem Let E1 and E2 be two measurable sets, then E1 ⋃E2 is measurable.
WE have two measurable set say E1 and E2.
we have to prove that union of these two measurable sets i.e. E1 U E2 is again measurable .
to prove that the union is measurable it is sufficient to prove that one inequality
lets go for the proof
using the definition of measurable set on E1 for any set say E1
and then using the definition of measurable set on E2 for any set say A-E1 Compliment
and then putting in each other and by applying countably sub additive and set theory identities

outer measure is countably subadditive :-

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[alfakarenmartinezhernandez]

the set E1 intersection E2 is
also measurable (why?)