I've been doing some exercices on Lebesgue integrations,and I had an idea of a 'theorem'.I would like to know whether it is true,and if yes,if my reasoning is sound.
Proposition:Let E be the unit interval and f a mesurable real valued function on E.Let x be some real number strictly between 0 and 1.Then if the the lebesgue integral of f is 0 on every measurable set of measure x,then f is almost surely equals to 0
My reasoning went as follows: Wlog,let us assume that the measure of the set where f is strictly positive is >0,I shall denote this set by P.Then,we can find a set T such that T='a 'small' part of P ' union 'a part of N or O' (where N stands for set of y st f(y)<0 and O for those st f(y)=0 (and I don't want to get to all the details,forgive me for using 'small') and the measure of T is x.Now,it's easy to see that either the measure of N or O have strictly positive value.Then we can construct T' of measure x st we replace 'small' part of P by a 'small' part of N union O.But this means that the value of Lebesgue integral should be different on those 2 sets,but they should be equal by the assumption -> contradiction.
Is this correct or there's a flaw in my reasoning ? Thanks in advance and please forgive me for not using mathematical symbols,I don't know how to write in Latex yet.