<p>In this paper, we establish that a norm bounded set <i>A</i> in a Banach lattice <i>E</i> is an order bounded subset of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(E^a\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>E</mi> <mi>a</mi> </msup> </math></EquationSource> </InlineEquation> if and only if every disjoint sequence in its solid hull is <i>ru</i>-convergent to zero. Based on this result, we define a quantitative measure <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\delta (\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and show that for every norm bounded set <i>A</i> in <i>E</i>, <i>A</i> is order bounded in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(E^a\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>E</mi> <mi>a</mi> </msup> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\delta (A)=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. As applications, we investigate the order boundedness of sets in atomic order continuous Banach lattices, and provide several necessary and sufficient conditions for an order continuous normed Riesz space to be a Banach lattice. In addition, we also obtain several sufficient conditions for a set to be <i>b</i>-order bounded in Dedekind complete Riesz spaces.</p>