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We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described here), and we can make $A$ have arbitrarily small measure on that one interval, which is what bof refers to in his comment. Now, by repeating those small-measure sets or large-measure sets as desired on successive intervals, we can control the asymptotic measure as in your limit so as to realize any desired value or a nonconvergent value.

We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described here), and we can make $A$ have arbitrarily small measure on that one interval, which is what bof refers to in his comment.

By repeating those small-measure sets on successive intervals, or by using large-measure sets as desired on successive intervals, we can control the limit value of your asyptotic measure so as to realize any desired value or a nonconvergent value.

We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described here), and we can make $A$ have arbitrarily small measure on that one interval. Now, by repeating those small-measure sets or large-measure sets as desired on successive intervals, we can control the asymptotic measure as in your limit so as to realize any desired value or a nonconvergent value.

We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described here), and we can make $A$ have arbitrarily small measure on that one interval, which is what bof refers to in his comment. Now, by repeating those small-measure sets or large-measure sets as desired on successive intervals, we can control the asymptotic measure as in your limit so as to realize any desired value or a nonconvergent value.

We can make those limits be any two numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction, and we can make $A$ have arbitrarily small measure on that one interval. Now, by repeating those small-measure sets or large-measure sets as desired, we can control the asymptotic measure of your limit so as to realize any desired value or a nonconvergent value.

We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described here), and we can make $A$ have arbitrarily small measure on that one interval. Now, by repeating those small-measure sets or large-measure sets as desired on successive intervals, we can control the asymptotic measure as in your limit so as to realize any desired value or a nonconvergent value.