New posts in Lebesgue Measure

Doubt about Exercise 5, Chapter 1, Shakarchi [RA]

If $\int_0^1f^k(x)\;dx\le M$, then $m(\{x\in[0,1]:f(x)>1\})=0$.

If $f∈C^1$ and $\{∇f=0\}$ has Lebesgue measure $0$, then $\{f∈B\}$ has Lebesgue measure $0$ for all Borel measurable $B⊆ℝ$ with Lebesgue measure $0$

$ \mu(|f|\geq \alpha) = \frac{1}{\alpha} ||f||_1$

Covering of Intervals

Question about Vitali Covering

When does Brunn-Minkowski inequatily $(m(A+B))^{1/d} \geqslant (m(A))^{1/d} + (m(B))^{1/d}$ become equality?

Real world application of Lebesgue measure as opposed to Jordan measure

If $fg_n$ converges in $L^p$ for all $f\in L^p$, then $g_n$ converges in $L^\infty$.

Is it $\mu(dx)$ or $d\mu(x)$? or they are equal?