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bof
bof
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The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set alongalone is not that easy. The Vitali set does not provide the example asked for above.

The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.

The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set alone is not that easy. The Vitali set does not provide the example asked for above.

added 16 characters in body
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Ma Joad
Ma Joad
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The question is quite "simple". Let $\lambda$$\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda(E) +\lambda ([0,1]\backslash E) >1? $$$$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda(E) +\lambda ([0,1]\backslash E) =1$$\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.

The question is quite "simple". Let $\lambda$ denote the usual Lebesgue measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda(E) +\lambda ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda(E) +\lambda ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.

The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.

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Ma Joad
Ma Joad
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Summability issues of measure when we decompose a measurable set into two non-measurable parts

The question is quite "simple". Let $\lambda$ denote the usual Lebesgue measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda(E) +\lambda ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda(E) +\lambda ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.