Weak $L^{1}$ norms of random sums

Let $\left\{g_{j}\right\}$ denote a sequence of measurable functions on $\mathbf{R}^{n}$, and let $\left\Vert\cdot\right\Vert _{WL^{1}}$ denote the weak $L^{1}$ norm. It is shown that

where $\left\{\epsilon_{j}\right\}$ is a sequence of independent random variables taking on values $+1$ and $-1$ with equal probability. Moreover, it is shown that

The paper concludes by providing an example indicating that, if $\left\Vert g_{1}\right\Vert _{WL^{1}}$ $= \cdots = \left\Vert g_{N}\right\Vert _{WL^{1}} = 1$, then the estimate

is the best possible.